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•Entered according to the Act of Congress, in the year 1898, 

By F. W. EWALD & COMPANY, , 
In the Office of the Librarian of Congress, at Washington. 



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TMP96-0^4397 



GEOMETRICAL DRAWING-. 



1. Drawing is the art of representing objects by means 
of lines. The entire subject may be divided into two gene- 
ral classes; free-hand drawing, when no instruments 
of precision are used, and mechanical (me-chan'ic-al) 
drawing, in which, in order to obtain great accuracy, 
tools are employed for an exact representation of an object 
in its actual size or according to a given scale. These classes 
may again be subdivided in various ways, as, for example, 
free-hand drawing into sketching and designing, and 
mechanical drawing into architectural (ar-chi-tec'tur-al) 
drawing, geometrical (ge-o-met'ric-al) drawing, and 
machine drawing. 

While the final objects to be attained by a practice of these 
three kinds of drawing may differ considerably, they are all 
dependent on a correct use of the instruments and a thorough 
knowledge of geometrical principles. For this reason we 
shall first take up these two subjects. 

INSTRUMENTS AND DRAWING MATERIALS. 

2. While a draughtsman may find it convenient to pro- 
vide himself with a large number of drawing instruments, 
only those mentioned in this Paper are of vital importance 
and are ample in number for the satisfactory execution of 
all the drawings in this course. 

The drawing-board should be made of well-seasoned 
pine, free from knots, and must have a smooth surface. 
For the work in this course the size of the board is 16 by 21 
inches, and its thickness about f of an inch. Drawing- 
boards are made of different sizes and construction to suit 
the sizes of the drawings and the requirements of the 
draughtsman or designer. Boards should be barred or 
doweled at the ends to stiffen them and resist any tendency 



GEOMETRICAL DRAWING. 



to twist, as well as to afford a suitable edge for the working 
of the T square. Instead of this method, cross-bars on the 
back of the board are preferred by many draughtsmen. 
These serve for the twofold purpose of strengthening the 
board and raising it from the table, which facilitates the 

moving of it. The cross- 
bars or cleats are dovetailed 
into the board, and are ta- 
pered along their length so 
that they can be driven into 
place, as shown in Fig. 1. 
Fl *. i- The left-hand edge of the 

board should be carefully trued, and no other edge should 
be used for the head of the T square to rest against. 





Fig. 2 shows a drawing-board with cross-bars, and also illus- 
trates the position of the paper, T square, and triangle. 

3. The T square, shown in Fig. 3 and in position Jn 
Fig. 2, should only be used for drawing horizontal lines. It 



is usually made of mahogany or cherry-wood, and consists 



GEOMETRICAL DRAWING. 



of two parts, A, the blade, and B, the head. These two 
may be permanently attached to each other, or the head may 
be movable for drawing inclined lines. The resetting of it, 
however, takes time and often causes inaccuracy. The head 
of the T square is held against the left-hand edge of the 
board, as shown in Fig. 2, and the T square is moved with 
the left hand until the upper edge, D, of the blade A is 
very near to the point through which the line is to be 
drawn. All the lines a, drawn thus, will be horizontal and 
parallel if the student holds his pencil in a vertical position. 
Be sure to hold the head of the T square tightly against the 
edge of the board, and never stand a T square on the floor ; 
it should be hung up. Never use the edge of the blade for a 
cutting edge, nor the head of the T square for a hammer. 

4. The Triangles. These are used for drawing verti- 
cal lines and lines making 
definite angles with the hori- 
zontal. For this purpose tri- 
angles are divided into 45° 
and 60° triangles, shown in 
Fig. 4 and Fig. 5 respec- 
tively. The 45° triangle has 

two angles of 45° each and Fig. 4. fig. 

one of 90° ; the GO triangle has one angle of 60°, one of 30°, 
and one of 90°. To draw a vertical line, lay the triangle C 
against the blade of the T square so that a right angle is 
formed, as shown in Fig. 2. The T square is held tightly 
against the board with your left hand, and the triangle 
brought in position with your right. Then hold the blade 
and triangle lightly with your left hand, keeping them from 
slipping, and draw the line b with a pen or pencil held in 
your right hand. For drawing lines making angles of 30°, 
45°, or 60° with the blade of the T square, the triangles are 
placed in such a position as to have the required angles next 
to the upper edge of the blade. 

5. Through a given point, to draw a line 
parallel to a given line. Should the line be horizon- 




►METRIC \L hi; \wiv, 



tal or rertical tin I 
, triangl 1. hut it" the lin< 

has .in inclined direction tria 

<-nlv are utad.- u-<- of. Th( 

held in the position shown 
6, the long 

• the Long edge of the 
Both B and . I are moved 

until "in • ♦•(!«_:<• of .1 C 
with tli«' given line, /»' ie 
held down with the 1 « -ft hand 
. I i- moved along the long i 
/»' until the edge which we 
cident with the given line passes 
through the given point, when a line is drawn through that 

point. 

If a Dumber of lines are to be drawn parallel to the given 
line, B is held down as shown in the figure and .1 is moved 
along the 1mh--.mIl:.- of B. Should the lines occupy a Bpace 
in excess of the distance through which .1 can be moved, 
hold 1 down with your left hand and move /> with your 
right until the new position of the triangles will permit the 
drawing of the required lines. 




6. To dram lines pei> 
pendicular | per-pen-dic u 
lar) to ot hers n hicn are 
mil her horizon! al nor 
\ erf leal. 

This method is similar t<> 
the one described in Art . .">, 

with tl cception that the 

mn of .1 i- made 
incide with the line c or </. 
a- shown in Fig. ;. The 
dge of triangle l> is 
then laid against the long 
if .1. and /; i^ held 
down firmly. .1 i> then 




GEOMETRICAL DRAWING. 



moved along with the left hand into the position shown 
by the dotted lines, when e and / can be drawn along the 
edge mo. These lines will be perpendicular to c and d. 

Before beginning to draw the first plate the student should 
practise the handling of the T square and triangles, under- 
standing the scope of their usefulness, and their manipula- 
tion. 



7. The compasses (com'pass-es) with its accessories, 
the divider-points, pen, pencil, needle point, and lengthen- 
ing bar, as shown in Fig. 8, are, next to the instruments de- 
scribed, the most important tools of a draughtsman. 
Compasses are used for drawing circles or arcs, 
large or small, with pencil or ink; and if the steel 
points are inserted in the sockets of the legs of the 
compasses, as shown in Fig. 8, the instrument be- 
comes a pair of dividers, which are used for laying 
off distances or for dividing straight lines or circles 
into parts. 

The compasses, when used for drawing circles or 
g as dividers, should be held be- 
tween the thumb and forefinger 
of the right hand. When lay- 
ing off distances, or dividing- 
lines or circles, turn the di- 
viders alternately to the right 
and left, varying the distance 
between the points until the 
required distance is obtained. 
The points on the dividers 
should be very sharp, so as not 
to make holes in the paper, 
which, besides looking ugly, 
| are small receptacles for the 
ink. 




8. The pencil-leg: is used for drawing arcs, circles, etc. 
Be careful thafyou keep it exactly the same length as the 



\ 



GEOMETRICAL DRAWING. 



needle-point. This is accomplished by drawing the pencil 
out a little after each sharpening. The method of sharpen- 
ing the pencil will be treated in Article 17. 

9. The use of the inking-leg, as its name implies, is to 
repeat the pencil work in ink; the ink must be India ink 
and waterproof. On examining the pencil- as well as the 
inking-leg, you will find a joint in them, the purpose of which 
is to enable you to bend the leg at that point, so that the part 
which contains the ink (in the inking leg) may be kept perpen- 
dicular to the surface of the paper whilst describing a circle, 
as is shown in Fig. 9. If the inking-leg would be kept as 




straight as the one carrying the needle-point, when the com- 
passes are opened to any extent only one of the nibs (the 
inner one) will touch the paper, and thus the outer edge of 
the circle drawn will be ragged and rough. In drawing 
circles be careful to bear as lightly as possible on the part 
carrying the needle-point, so that your center is not pricked 
through the paper; for then, as eaeli concentric circle i^ 
drawn, the hole will become larger, until all chance of fol- 
lowing the exact curve will be lost, and when yon come to 



GEOMETRICAL DRAWING. 



ink the drawing you will find the difficulty still further 
increased. 

10. When it is required to draw a circle with a larger 
radius than could be described with the compasses in their 
usual form, a lengthening* (length'en-ing) bar is used; 
this is an extra extension rod, which fits into the socket of 
the leg of the compasses, and has at its other end a socket 
into which the end of the pencil-leg or inking-leg fits. This 
forms a pair of compasses with one leg very much longer 
than the other, and which is, therefore, rather awkward to 
manage. Here again the student is reminded that the pen- 
cil-leg and inking-leg must be bent at the joint, so that they 
may be perpendicular to the surface of the paper. 

11. The following hints will be found useful: 

(1) See that the needle-points as well as the divider-points 
of your compasses are round, and not triangular, which 
latter form makes a hole much larger than the former 
does. 

(2) See that this point is not too fine; it should be rather a 
blunt point than otherwise, only just sharp enough to prevent 
it slipping away from the center. 

(3) Should either of these faults exist they may be easily 
remedied by drawing the point a few times over an oil-stone, 
remembering to keep turning it around whilst moving it 
along. 

(4) If the joint at the top of the compasses is troublesome to 
open, or it works too freely, it should be adjusted by means of 
the accompanying key until it works easily and can be manip- 
ulated with one hand only. The joints in the pen- and pencil - 
legs should be adjusted by means of the screwdriver, which 
is part of the key. Be sure that the joints do not open or 
close by the weight of the parts, as this action will open or 
close your compasses, and the circumference of the circle 
which you are drawing will not be continuous. 

(5) Learn at the outset how to open and close the com- 
passes with one hand, as it is unnecessary and looks awkward 
to use both hands for the handling of so small and delicate 



10 



GEOMETRICAL DRAWING. 



a tool as a pair of compasses. A draughtsman is judged as 
much by the condition and handling of his instruments as he 
is by the work which he performs with them. 



12. The compasses are, however, not well adapted for 
drawing small circles and arcs, on account of their size and 
weight. For this work, as well as for the drawing of a 

number of circles of the same 
size, the bow-pencil and bow- 
pen, shown in Fig. 10, are used. 
The legs of these instruments, 
instead of being united by a 
hinge joint, are made in one 
piece so as to form a spring, which 
by its action tends to force the 
points apart; they are acted upon 
by a nut, which, screwing upon 
a bar fixed in one leg and passing 
through the other, as shown in 
Fig. 10, enables very delicate ad- 
justments to be made. The two 
legs of the instrument should be 
adjusted to very nearly the same 
length, the needle-point project- 
ing a trifle beyond the pen- or 
pencil-point so that very small 
circles may be drawn. 
To open or close either one of 
the instruments, put the needle-point in position and press 
down slightly on the top of it with the forefinger of the right 
hand. Then turn the adjusting nut with the thumb and 
middle finger of the same hand. 

13. The most frequently used, and, we might also say, 
misused, instrument is the drawing-pen, or, as it is some- 
times called, the ruling- or right-line pen, shown in Fig. 11. 
It is similar to the inking-leg of the compasses just described, 
and is used for inking in straight lines and irregular curves. 
It consists of the nibs, the handle, which is made of ivory 




GEOMETRICAL DRAWING. 



11 



or ebony, and a needle-point which is attached to the 
upper part of the handle and is screwed into the lower 
part of the blades. The pen should be held nearly 
upright, with the forefinger resting on the head of 
the screw, as shown in Fig. 12. This is a natural po- 




sition, and the student will have no difficulty in holding the 
pen in the above manner after a little practice. Do not rest 
the wrist on the board or T square, but hold it in the position 
shown in Fig. 12, from which it will be seen that only two 
fingers move along the blade of the T square. 



14. In order to draw a smooth, even line, the student 
should observe three things : 

(1) Hold the pen perpendicular to the board, so that both 
nibs touch the paper. If one blade only rests on the paper 
the line will be ragged. 

(2) Do not press the inner blade against the edge of the 
T square, as this will cause the two blades to close up and 



12 GEOMETRICAL DRAWING. 

the line will be thinned at that point. The edge of the 
T square should be used as a guiding edge only. 

(3) The ink must flow freely. If the ink has become dry. 
wipe it off with cloth and put new ink into the pen. or run a 
piece of paper between the blades. It is always safe, after 
a few moments delay, to try the pen on another piece of 
paper before a line is made on a drawing. 

Never put a drawing-pen away without first cleaning it 
thoroughly. To accomplish this conveniently the set screw 
which adjusts the width of the lines should be taken out and 
the blades be separated. For this purpose there is a joint at 
the bottom of the blades. 

The ink should be placed between the nibs with a quill or 
pen : under no circumstances should the pen be dipped 
into the ink. Be sure to wipe off all the ink from the outside 
of the blades, for if there is any ink on the blade which rests 
against the edge of the T square the student is almost sure to 
make a blot. 

15. Drawing-Ink. The best India ink should be 

used for inking in a drawing, and "Higgins* Waterproof 

Liquid India Ink," shown in Fig. 13, is recommended to our 

a students. To the cork of every bottle of this 

X ink a quill is attached, which should be used 

Jg^ for rilling the pen. This is done by dipping 

^ the quill into the ink and then inserting it 

between the blades of the drawing-pen. Too 

much ink in the pen is liable to drop, and the 

I pen when refilled should not contain more ink 

| than will occupy the space of a quarter of an 

I inch along the blades. Again, be sure 

not to have any ink on the outside 

fig. 13. of the blades. Do not use common writing 

ink. as it corrodes the drawing-pens and does not dry as 

quickly as India ink. Drawings on which writing ink has 

been used will not be accepted. 

The quick-drying quality of liquid India ink will annoy 
the student considerably, as the ink will dry in the pen in a 
very brief time and refuse t<> flow. This is especially the 




GEOMETRICAL DRAWING. 13 

case when fine lines are being drawn, and the only remedy 
is to clean the pen frequently and then refill it. If you want 
to stop working for a while, or have completed an evening's 
work, be sure to clean your pen thoroughly and open the 
blades so as to remove the strain on them. Never use a knife 
for scraping off the hardened ink, and if you use water be 
sure to ' wipe off all the moisture before the pen is put into 
the case. It often happens that the ink has dried only at 
the extreme points of the blades ; all that is required to start 
it flowing is to wet the end of the finger and apply it to the 
points of the pen. To avoid the spilling of the ink, and to pre- 
vent its drying up or filling up with dust, it is essential that 
you keep the bottle corked whenever you are not filling 
your pen. 

If the ink should require to be thinned or diluted, use a 
mixture of clear water and ammonia — four drops of ammo- 
nia to the ounce of water. 

16. Drawing-Paper and how to fasten it to 
the hoard. For the drawings in this course the student 
requires Whatman's hot-pressed drawing-paper, demy size, 
which is 15 // x20 // . This paper is of excellent quality, well 
adapted for ink work, and will withstand considerable eras- 
ing. Four thumb-tacks are used for securing the paper to 
the drawing-board, one at each corner of the sheet. The 
first thumb-tack inserted is the one marked c in Fig. 2, about 
Y from the upper and left-hand edge of the paper. The 
T square is then placed near the top edge of the paper, 
which is moved until the top edge is parallel with the blade 
of the T square. The thumb-tack marked d is then inserted, 
the paper being stretched diagonally across the board. 
Thumb-tacks e are then pushed in, and you are ready to 
begin your drawing. 

17. Drawing-Pencils and how to sharpen 
them. As a rule hard pencils are not the best for mechani- 
cal drawings which are to be inked, as they are liable to 
make grooves in the paper, the bottom of which the nib of 
the drawing-pen does not touch, and hence the edges of the 



14 



GEOMETRICAL DRAWING. 




line will be ragged ; and, further, lines which are drawn 
with very hard pencils are difficult to rub out. We recom- 
mend the use of a No. 4 Dixon's pencil, 
marked H, which indicates its hardness. 
For mechanical drawing it is best to have 
a flat point on the pencil, as shown in 
Fig. 14. This is done by cutting away 
the wood and leaving about £ of an 
inch of lead projecting, which is then to 
be cut until it is thinned to a flat, broad 
point like a chisel, with the lower edge 
slightly rounded. The broad side of this 
point is moved along the edge of the 
T square or triangle, and the line thus 
drawn will be found to be much finer than 
one drawn with a round point. The 
chisel point is economical in various ways, for it will not 
break easily, and the point once cut can be rubbed from 
time to time on a fine piece of sandpaper or file, or even the 
edge of the drawing-paper. The lead in the compasses 
should be sharpened in a similar manner, but with a nar- 
rower edge, and should be so inserted in the pencil-leg that 
only one line is drawn with the same radius and center 
whether the compasses be moved in one direction or the 
other. The student should learn to make light lines and 
under no circumstances wet the point of his pencil. 

18. Erasers (e-rasers) and how to clean a draw- 
ing. To the beginner, a pencil- and ink-eraser are tools 
with which he cannot very well dispense, but their incorrect 
or excessive use will ofttimes disfigure an otherwise credita- 
ble drawing. The less either one of the erasers is used the 
better it will be for the paper, and the pencil eraser should 
not be used until the entire drawing, with the exception of the 
hatched portions, letters, and figures, is inked in. Then rub 
out as lightly as possible all pencil lines which have not been 
inked over and any other disfigurements on the paper. For 
erasing ink lines a sharp knife blade may be used instead oi' 
the ink eraser. In either case the paper shouldjje smoothed 



GEOMETRICAL DRAWING. 15 

down with the back of the finger nail or a bone handle, if 
ink is to be applied at that particular spot. Remember in 
all your work that prevention is better than cure, and it is 
more advisable to do your work slowly and with care than 
to do it quickly and disfigure the drawing by using the era- 
sers excessively. 

19. The draughtsman's scale, unlike the ordinary two- 
foot rule, is made of boxwood and has its divisions marked 
on beveled edges, so that measurements may be made by 
applying the scale directly to the drawing. Its divisions do 
not begin at the two ends, but some distance away, so as to 
avoid any error which might be introduced by the wearing 
away of the ends of the scale. The scales are either flat or 
triangular, but there should never be more than two scales 
on each edge of the instrument, one at each end. Drawings 
are either made full size or to a scale. The working- 
drawings of details should be made to as large a scale as 
convenient, and, if possible, should be full size. The smaller 
scales are used for the general views of large objects. For 
full-size drawings the edge marked A (Fig. 15) should be 
used, which is divided into twelve inch-divisions, which are 
subdivided into halves, quarters, eighths, and sixteenths. 

20. Suppose it is required to make a drawing \ size, or 
to have 3 inches on the drawing represent 1 foot on the 
object. If we then lay off 3 inches and let those represent 
1 ft., then y^ of these 3 inches, or \ of an inch, will represent 
1 inch on the object. Such a scale is laid off at B, and 
should be used in the following way : 

Suppose we wish to lay off a distance on the drawing equal 
to 2' — 5i" on a scale of 3 inches to one foot, or a quarter 
scale, we begin at the zero mark on the side marked B (Fig. 
15) and lay off the distance which corresponds to 2' on one 
side of it and the 5-J-" on the other. For this purpose the zero 
mark is not placed at the end of the scale, but a distance 
from the end which represents one foot. A scale of 1^ 
inches to one foot, or an eighth scale, is laid off at the end 
marked C, and this is used in a similar manner as the one 



10 



GEOMETRICAL DRAWING. 



quarter or any other scale. If it is required 
to lay off 4' — 7i", we lay off 4 ft. (represented by 
6 inches on our scale) on one side of the zero 
mark, and 7i inches on the other, as is shown 

§rh ij in Fi s- 15 - 

After the student understands this principle 
and has practised the use of scales for a long 
time, he is in a position to construct a scale for 
a particular purpose at any time. 

If, for example, a one-twelfth scale, or a 
scale of 1 inch to one foot, is wanted, the stu- 
dent can lay off 1 inch on a strip of paper and 
divide that into 12 parts. Then each one of 
these 12 parts represents one inch on the ob- 
ject. These can again be subdivided into 
halves and quarters if it is found necessary. 

For a sixteenth scale, or a scale of f inch 
to one foot, it is only necessary to let the 14 
inches on the eighth scale represent two feet 
on the object; and for a twenty -fourth scale, 
or a scale of -| inch to one foot, halve each 
dimension on the scale of 1 inch to one foot. 

The above represent the most common scales 
used in practice, and a correct understanding 
of them will enable the student to make a 
drawing to any desired scale. 

21. Irregular (ir-reg'u-lar) curves, or. 

as they are sometimes called, " French curves " 
or "sweeps,'' are rulers of irregular shapes 
used for drawing curves other than 
arcs of circles. The curve should be 
so constructed (see Fig. 16) that by 
means of it a curve of any shape can 
be drawn. It is used in the following 
wav: A certain number of points have 
been determined through which a curve 
is to be drawn. Find some part on your ir- 
regular curve which will pass through at least 



GEOMETRICAL DRAWING. 



17 



three of the points, and draw the curve through them, taking 
care, however, that the termination has such a direction that 
the next part of the curve joined to it will not form an un- 
sightly point at the junction. Also be sure that the line does 
not curve out between the points, or, perhaps, has a tendency 
to curve in. In placing the curve in a new position, let the 
last part of the curve just drawn be retraced by the first 
part of the curve in the new position. It is advisable, espe- 
cially if the points are a considerable distance apart, to sketch 
the curve free-hand and then to let the penciled curve act as 
a guide for the irregular curve. All curves should be pen- 
ciled in before they are inked. 

We will illustrate the method of using the irregular curve 
by an example. Let it be required to draw a curved line 
through the points a, b, c, d, etc. (Fig. 17). 




We first place the curve in the position marked A, so^that 
part of it joins the points a, b, and c. We draw this section 
a, b, c, or rather stop at some place between b and c, so that 
there will be no angle where the next part of the curve will 
join it. We then place the curve in the position B, shown 



18 



GEOMETRICAL DRAWING. 



dotted in Fig. 17, and find that it retraces part of the distance 
be, and joins the points c, d, e, and/. We draw this section 
of the curve to some point between e and /, and place the 
curve in the position C. Here we again retrace part of ef, 
and complete the curve by joining /, g, h, i. It will be 
noticed that the curvature changes at the point /, and at such 
a point the curve may be joined without retracing a section 
of it on either side. 

The student should practise the joining of points with 
irregular curves, first sketching them free-hand, then pencil- 
ing them by adjusting the irregular curve to the free-hand 
curve, and, lastly, inking them, taking care to hold the pen 
close to the curve and always in the direction of the curva- 
ture. 

22. The protractor (pro-tract'or), one form of which 
is illustrated in Fig. 18, is an instrument used for laying off 
or measuring angles and dividing circles into any number of 
equal parts, or laying off degrees on the circumference. Pro- 




tractors are made either of metal, paper, or some transparent 

material. The semicircle which. is its one boundary, the di- 
ameter of the circle with the marked center being the other, 
is divided into 360 parts, each part being one-half of one 
degree. It is numbered from a to h and /> to a from t) to 180 



GEOMETRICAL DRAWING. 



19 



To use the protractor for measuring or laying off 
angles we proceed as follows : 

One side of the angle which we intend to lay off has the 
direction a b (Fig. 18), and the vertex is located at the point 
o. Making the lower edge of the protractor, or rather the 
line passing through the 0° and 180° mark, coincide with line 
ab, or ab prolonged, of the proposed angle, and placing the 
center, o, of the protractor over the point which is to be the 
vertex of the angle, we lay off the required number of de- 




grees, as, for example, 62° shown in the figure. A mark is 
then made at this place with a sharp-pointed pencil, and, 
after the protractor has been removed, the mark is joined to 
the vertex o. It is advisable for greater accuracy to prolong 
the line ab so that both the 0° and the 180° mark will coin- 
cide with the line. 

For dividing the circumference of a circle into any number 
of equal parts, say 30 (see Fig. 19), the point o on the pro- 



20 GEOMETRICAL DRAWING. 

tractor is placed directly over the center of the circle, and the 
line joining the 0° and 180° marks is made to coincide with 
the diameter of the circle, or the diameter prolonged as is 
shown in the figure. The diameter ab is Jalso prolonged 
and will coincide with a line joining the center, o, and the 
90° mark on the protractor. We then lay off on the semi- 
circumference of the protractor 12 divisions, as 0-1, 1-2, 2-3, 
etc., giving us 15 parts. These points are then joined by 
lines to the center of the circle, as shown in Fig. 19, cutting 
the circumference of the circle in the points 0', 1', 2', 3', etc. 

23. In conclusion, the student is urged to remember that 
the mere possession of a case of instruments, however good, 
will not constitute a draughtsman. The instruments are 
merely the tools — the mechanical agents through which the 
mind acts; and it cannot be denied that the more thoroughly 
the mind comprehends the object to be drawn, the more willing 
and intelligent servants will the hands become, and the more 
accurately will they guide the compasses or the drawing- 
pen. The student will, no doubt, find it difficult at first to 
draw very fine lines, or to get them to intersect each other 
exactly as required, especially if he has been engaged in 
some hard manual occupation during the day; but he will 
find a little practice will soon overcome this, if he but starts 
with patience, energy, and the earnest desire to excel. 

LETTERING THE DRAWINGS. 

24. Lettering may be divided into two general classes, 
namely, mechanical and free-hand. These are again 
divided into different styles of letters, and the proper choice 
of class and style, as well as their correct execution, is one of 
the most difficult problems for the beginner. However, draw- 
ing-office experience has shown that the free-hand letter and 
the style shown below should be used for all lettering on a 
mechanical drawing, except the title, which should be made 
with the T square and triangles, and the style adopted should 
be the one called block lettering. 

The student must be told at the outset that unsightly 



GEOMETRICAL DRAWING. 21 

lettering* will not only spoil an otherwise excellent drawing, 
but will also confuse and discourage the workman who is 
obliged to use the drawing. A draughtsman should acquire 
a definite style of lettering and use this throughout his work. 
Ordinary script or poorly executed lettering will not be 
accepted, and the student will find the time he devotes to 
practising how to letter profitably spent. 

He should do his lettering without haste, and not be 
discouraged if he finds that the lettering of a drawing con- 
sumes more time than the making of the drawing itself. 
This is often the case on drawings of details, and even the 
most experienced draughtsman will devote all the time that 
is required to the careful execution of his letters and figures. 

The style of lettering which we have adopted, and which 
is shown in Fig. 21, has been approved by the leading manu- 
facturing concerns in this country, and we have found that 
any student who does his work conscientiously can easily re- 
produce these letters and do them neatly and rapidly without 
the use of instruments. For lettering, a Gillott's No. 303 
writing-pen should be used, such as is shown in Fig. 20. 
The height of the capital letters is ^^ m 



- ■ 



-f Y m - , an d that of the small letters is *^~~: 7- -t M5Si 
f of this, or ^ in. They should not 
be any larger or smaller than this. 



ABODE FGJ^IJJiTEMJVOFQRS TUVWX^FZd 
a be defghijJtlmnopqrstuvwxyz, /Z34567890 

/Z34567890, GEAR. Furnace Door Scale §=/ Ft 



25. After the student has decided what lettering ought 
to be placed on his drawing, and where it should be placed 
(two questions of great importance), he draws the guide lines 
for the tops and bottoms of the letters with a T square, the 
required distance apart, as stated above. These lines should 
be penciled as lightly as possible, since they are to be erased 



22 GEOMETRICAL DRAWING. 

after the lettering has been inked in. Be sure not to have 
the tops and bottoms of the letters extend beyond or fall 
short of the guide lines. If the student fails to observe this 
point the lettering will look like the second word * ' Geome- 
trical " in Fig. 21. 

26. Another important point to be observed is to give all 
letters the same inclination. 

As will be seen by inspecting the letters of the words 
" John Lamp" following the word " Geometrical " in Fig. 21, 
their sides all have the same slant, which should be 60°, as is 
indicated by the dotted lines. The two sides of the letters N } 
M, n, u, etc., are parallel and have the common slant of all 
the letters. At the beginning the student is advised to 
draw a number of parallel slant lines, which will aid him in 
keeping the inclination uniform. 

27. Each letter, or rather set of letters, in this system 
has its own particular construction, and this should be care- 
fully studied by the student. For convenience of study, and 
on account of the principles involved in their construction, 
we will divide the capital letters into four sets, as is shown 
in Fig. 22. 



&FW1LT £¥p& b f¥zJf?M BDJKPR 000&M 

A a be 6d c ab^ abc/ddb ^ */> * 

ll,7ri7n, oo, pp, rr, s j, t I, n Tin, c c, g j p, ace, d, b, 
' M$ $*$&$&$& 7 £345678 90, 24-SzM 

Fig. 2-,'. 

In the first set, A 9 which includes the letters E, F. H, I, 
L, T, we deal simply with horizontal lines and lines having 
the common slant, and the main points to be observed aiv to 
have the inclination uniform and to have the horizontal 
lines coincide with the guide lines. The lines marked a in 
the letters E, F, and 11 are drawn midway between the 
upper and lower guide lines, and the left-band side in the 
letter H is slightly curved at the bottom. 



GEOMETRICAL DRAWING. 23 

In set I> we include those letters which are constructed 
about a center line having the inclination of the common 
slant line. The student should make the points a and b in 
these letters, and a and g in the W, an equal distance from 
the center lines, and in A and Fthis distance is to be -fe in., 
making the width of these letters $% in. In both letters the 
student should first draw the common slant and then draw 
the sides, as explained above. It will be seen that the side 
cb in the letter A, and ac in V, are nearly perpendicular to 
the guide lines. 

To make any letter in this set, the common slant line, 
which is also the center line, must first be drawn. The 
upper part of the T begins a little below the center, and the 
points a and b are a little less than -^ in . distant from the 
center line, so as to make the width of the letter -£± in. at the 
top. 

The student should not attempt to make the U with one 
stroke of the pencil or pen. First make the line ac, begin- 
ning at a, and then draw be, beginning at b. 

The sides ac and bd of the letter M are parallel, have the 
common slant, and ac is curved a trifle at the bottom, as 
shown. The letter is 3 3 ¥ in. wide, and is, therefore, as wide 
as it is high. The student should first draw the two sides ac 
and bd. Then draw the line ef half-way between them, 
and join a and b with /. The lines af and bf are slightly 
curved. 

The letter W is composed of two V% each narrower at the 
top than the letter V. The line eb is first drawn, having 
the common slant; then the points a and g are laid off fe in. 
from e, and the lines ab and gb are drawn. The part gdc 
is drawn in a similar manner. The entire letter is, therefore, 
i in. wide; ab and gd, as well as gb and cd, are parallel, 
and ab and gd are nearly perpendicular to the guide lines. 

The letters U, X, N, and Z are made f± in. wide, the X 
and Z being a trifle less than this at the top. The student 
should observe that the small horizontal lines in which some 
of the sides of the letters terminate should be horizontal and 
exactly in line with the guide lines. They should not be 
omitted from letters on which they belong, nor should they 



24 GEOMETRICAL DRAWING. 

be placed on letters where they do not belong. Let the 
student be guided by the letters as they are shown in Fig. 22. 

The letters in the set C are combinations of straight and 
curved lines, and the main feature to be observed here is the 
fulness of the curves and the proper width and slant of the 
letters. 

In the set marked JD in Fig. 22, those letters are shown 
which are composed entirely of curves, namely, C, G, O, Q, 
and S. 

The general inclination of these letters is the same as for 
all the others, and the student should make a line having 
the common slant, to guide him in the construction of these 
letters. The C and G can be made with one stroke of the 
pencil or pen. The O and Q should be made in two strokes, 
the first from a to & on the left of the common slant line, and 
the second from a to b on the right of it. The S can be made 
in one stroke, from b to c. The student should draw the 
two slant lines ac and bd, and have the letter S touch these 
lines at a, d, and c, and not at b. As a guide for the middle 
portion of the letter, the line ef is drawn, which should not 
have a greater slant than is shown in the figure. 

28. Again referring to Fig. 21, the student will notice 
that the b, d, h, i, j, k, I, and p have sharp corners at the 
top, and that the q is sharp at the bottom ; all of them ter- 
minate in a small horizontal line which coincides with the 
upper guide line in all the letters named except the q. All 
small letters have the common slant and should be con- 
structed carefully and in strict accordance with the copy in 
Fig. 21. The first letter of each group in the middle line of 
Fig. 22 is printed correctly, and the ones following show a 
construction which should be avoided b}^ the student. 

The letter g should be constructed as shown, the upper and 
lower loop touching the common slant line, which should be 
drawn for a guide by the beginner. The s is printed similar 
to the capital S. The d should be made in two strokes and 
the b in one. 

In conclusion, we want to say that all lettering should first 
be done with pencil and then with pen and ink. The student 
should do his lettering slowly, and practise the style and con- 



GEOMETRICAL DRAWING. 25 

struction of the letters on a sheet of paper before he attempts 
to put them on his drawing. The letters should be spaced 
evenly and should all have the same inclination. 

29. Again referring to Fig. 22, the student should study 
the construction of the numbers and make them exactly like 
the copy shown there. The workman in the shop, as well as 
the builder, is guided in the construction of apparatus or 
dwellings by the dimensions which he finds on the drawing. 
It is, therefore, of the utmost importance that the numbers 
indicating the size of a particular part on a drawing should 
be plainly written. Adopt the style shown in Figs. 21 and 
22. They all have the same general slant as the letters and 
should be constructed as follows : 

The first set of numbers in the last line of Fig. 22 shows 
the figures as they should be printed. The student will ob- 
serve that all the numbers except the 1 and 4 are constructed 
about a center line which has the common slant (60°), and 
that the curves in the numbers 2, 3, 5, 6, 8, 9, are full and 
round. The 1 is made in one stroke. The 2 touches the 
two parallel slant lines at four points. The 3 touches them 
only at three points, as shown. The line ab in 4 is drawn 
from a point a located at about ^ of the distance between 
c and d. The 5 has three points in contact with the slant 
lines, and the line ab is straight and coincident with the 
upper guide line. The 6 touches the slant lines at three 
points and should be drawn in two strokes — the first from a 
to d, and the second from b to c and joining the first curve 
at d. The point c in the 7 should fall midway between the 
point a and the center line. The line ab should be straight 
and be curved. The 8 should touch the slant lines at two 
points, and the upper loop should be slightly smaller than the 
lower. It should be made in two strokes, the first being the 
same as the letter s, that is, b, a, d, c. The second stroke 
is from b to c, completing the number. The 9 touches the 
slant lines at three points, and the upper loop extends a little 
below the center of the two outside guide lines. This num- 
ber is made by beginning at some point between b and d 
and drawing the loop dab; then, beginning at the first 
starting point, the rest of the number is drawn. The back of 



26 GEOMETRICAL DRAWING. 

the numbers 6 and 9 should be curved lines and only touch 
the slant line at one point marked b. The is made in two 
parts, beginning each part at a and joining them at b. 

The second set shows the numbers as they should not be 
made, and the student should avoid these mistakes : Do not 
affix a small line to the 1, and be sure to give it the common 
slant. Do not make the lower part of the 2 too large. Do not 
cramp the numbers, as shown in 3, 4, 8, 9, and 0, but make 
the curves full and round. The horizontal line in the 4 
should be drawn a trifle below the center of the two outside 
guide lines. The lower loop of the 5, and the upper loop of 
ths 6, should not protrude as they do in these specimens. 
The line connecting the two outer guide lines in the 7 should 
not be straight, but should have a double curve, as shown in 
the first set. 

The height of the numbers is ^ in., the same as the capi- 
tal letters, and in case we have a fraction, as is shown in 
Fig. 22, the total height of the fraction is ^V i n -? each num- 
ber being a little less than T \ in. high. 

30. For the titles of all the plates and for large head- 
ings, the block letters, shown in Fig. 23, are used. 

Unomgrstuv 



oqn _ 13579 ie i rr 

03U, 2 4 6810 I02 



The letters are mule with T square, triangles, and com- 
passes, and the only free-hand work about them is the fill- 
ing in of the spaces between the lines made with the draw- 



GEOMETRICAL DRAWING. 27 

ing-pen. An ordinary writing-pen may be used for this 
purpose. 

The student first draws two horizontal lines -^ in. apart ; 
this space he divides into five spaces of -^ in. each, which 
gives him the guide lines for the tops, bottoms, and centers 
of the letters. He then draws two vertical lines -^ in. apart 
(see a and b in the figure) ; i in. from b he draws the verti- 
cal line d, and T V in. from this the line e, making the total 
width of the letter \ inch. This is the width of all the let- 
ters, with the exception of the M, which is T 5 ¥ in. wide; the 
W, which is -J^ in. wide; the I, which is -^ in. wide; the A, 
which is -3 9 F in. wide; and the J, which is ^ in. wide. The 
distance between any two letters is xV m -> except where P 
and F precede A ; where J follows F, P, T, V, W, or Y ; 
where V, W, or Y follow L ; and where A is adjacent to T, 
V, W, or Y. In these cases the lower extremity of one 
letter is in the same vertical line with the upper extremity of 
the other. 

After the student has laid out his horizontal and vertical 
guide lines which give him the width, height, and thickness 
of the letters, as well as the spaces between them, he will 
find the remaining work very simple. However, we will 
point out a few details which deserve special attention. 
The parts marked a in the letters C, D, G, O, and Q are flat, 
this being due to the fact that the letters are higher than 
they are wide. In the letters A, V, W, and Y draw a 
line midway between the outside vertical lines, which is 
marked c in the letter A. On each side of this line lay off 
sV in., which gives a width of -^ in. to the tops and bottoms 
and very nearly T y to the stems of these letters. The P, R, 
B, D, and S are flat at the top, and the last three also on the 
bottom. To construct the W, first draw two vertical lines, 
ag and fh, \\ in. apart. Midway between these draw the 
line iv and lay off the points c and d -£% in. on each side of 
i. Midway between ad and cf draw the lines o, n, and 
measure off ^ in. on each side of the points n. Then com- 
plete the W by joining these points with a, b, c, d, e, 
and/. 

To construct the M, draw the two vertical lines ac and de 



GEOMETRICAL DRAWING. 

3 V in. apart, and midway between these draw a line, fa in. on 
each side of this line, on the lower guide line la}" off two 
points and join these with the points a and d. From b and 
/ draw lines parallel to these, and then complete the letter 
by drawing the vertical and horizontal lines with T square 
and triangle. 

The three straight horizontal portions of the letter S are 
joined on the upper left and lower right by semicircles, and 
at the upper right and lower left quadrants form the ends of 
the letter. 

After the vertical lines of the letter K have been drawn, 
the line ab is drawn from the point a, located on the first 
horizontal division line from the bottom, to b, the upper 
right-hand corner of the letter, c is located midway between 
the lines ef and bd on the second horizontal division line 
from the top. d is located vertically below b. 

The width of the letters E and F is four spaces, as is the 
case with all the letters except those mentioned above. The 
points b in E and F are located midway between the outer 
extremities of the letters, or two and a half spaces to the 
right of the first vertical line. 

After these few explanatory remarks the student should 
have no difficulty in constructing the other letters by having 
frequent reference to Fig. 23. 

31. The height and width of the numbers in this system 
of lettering are the same as those of the letters, namely, -fe 
in. and J in. respectively. 

The construction of the numbers should require no further 
explanation after the student has mastered the construction 
of the letters. He should frequently refer to Fig. ft 3 and 
diligently study each number. Carefully observe the curva- 
ture of the numbers 2 and 7, and in the number 5 notice that 
the point a is located 2 J spaces from the top of the number. 

The total height of the fractions is § in. , and the height of 
each number fa in. The width of these numbers is 1 in. 

The "and" sign and punctuation marks should also be 
carefully studied and practised. 

In filling in the spaces between the lines made with the 



GEOMETRICAL DRAWING. 29 

drawing-pen, use a pen that does not scratch or tear up the 
fibres of your paper; and use your drawing-ink for this pur- 
pose, not plain writing-ink. 

HOW TO SEKD IN YOUR WORK. 

32. This course consists of eight plates, five on the solu- 
tion of geometrical problems, two on projection 

(pro-jec'tion), and one on intersections (in-ter-sec'tions) 
and developments (de-vel'op-ments). 

The first five plates are to be drawn from the instructions 
in this Paper, and no copies will be sent to the student for 
reference, except a copy of the first plate, which is annexed 
to this Instruction Paper. 

Copies of the remaining plates will be sent to you as you 
need them. As spon as you have finished your drawing of 
Plate I., send it to us at once in the mailing tube which we 
sent to you. Do not wait until you have several plates 
finished before you send them to us, but mail each one as 
you finish it. 

After you have sent Plate I. to us, read the instructions for 
drawing Plate II. and begin to work on that plate. Our 
instructors have in the meantime corrected your first plate 
and have returned it to you. The corrections made by them 
on the drawing or on a separate sheet of paper should be 
carefully noted by you, and should guide you in the execution 
of the remaining plates. Not until the first plate has been 
returned to you should you send the second plate to us, for 
we are desirous of having the mistakes on the first plate 
corrected on the second and succeeding plates. After you 
have sent the second plate to us for corrections begin to 
draw the third plate, but do not send this to us until Plate II. 
has been returned to you. This method is to be followed 
until every plate in the course has been drawn and criticised. 

Do not fold your drawings or send them to us in anything 
but a mailing tube such as we furnish to our students. 
Write your name and address in full with lead pencil on the 
back of each plate. This will facilitate matters at our 
schools and will insure the quick return of your work. 



30 GEOMETRICAL DRAWING. 

Do your work neatly and keep your drawings clean. Trim 
them carefully and avoid the punching of holes into them 
with compasses or dividers. Work slowly and stud}' the 
principles involved in each new plate. Should you have any 
difficulties at any stage of the course, write to us at once and 
our instructors will clear them away. Do not feel discou- 
raged if your drawing is not as good as the copy we send you. 
or if it is returned to you with a large number of corrections. 
We do not expect perfect drawings from you at the begin- 
ning, and in pointing out your faults we are simply doing 
our duty, namely, to guide you in the right direction, which 
leads to a complete understanding of the subject and a 
knowledge of how to make geometrical and mechanical 
drawings. 

GENERAL DIRECTIONS FOR DRAWING THE 
PLATES. 

33. Next to knowing how to use the instruments and 
how to letter correctly, is a knowledge of how to properly 
locate the various views of a drawing on the sheet of paper. 
The views should be so located on the paper that there is no 
crowding of the parts. Every drawing has a cutting* 
edge, which is far enough away from the outside edge 
of the sheet of paper to allow for the cutting out of the 
thumbtack holes, and the student may use this margin for 
testing the flow of ink in his pen and the width of his lines. 
The size of the drawings in this course, after they have been 
trimmed along the cutting edge, is 14|" X IS". The work- 
ing edges (sometimes called border lines) within which the 
drawing is made are drawn V from each cutting edge, giving 
a space of 13£* X 17". On every drawing the cutting edges 
and border lines should be drawn first and the drawing then 
be spaced inside of the latter. In trimming your plates take 
care not to cut on either side of the cutting edges. 

The first five plates in this course will deal with practical 
geometrical problems, and the student will be required to 
draw these from the descriptions and illustrations given in the 
following pages. A copy of the first plate will be found in 



GEOMETKICAL DRAWING 31 

this Instruction Paper. The student should follow the 
explanations closely and learn the principles involved in the 
construction of each problem, so as to be able to apply them 
whenever he meets with a similar problem in his practice. 
In fact, the object of each plate in the course is twofold — 
namely, to teach the student how to draw, and, secondly, to 
teach him a new principle whenever he draws a new figure. 
The work should be done accurately and neatly, and be sure, 
therefore, that you do not begin to draw until you have 
washed your hands. All the work should be penciled in 
before you do any inking, and all curves should be inked 
before the straight lines; for it is easier to join a straight line 
to a curve than it is to have a curve join a straight line with- 
out causing an ugly place at the joint. As stated in Art. 
22, the student should do his lettering with great care. 
The drawing should not be lettered until all lines and curves 
have been inked in. You must not attempt to do the letter- 
ing without drawing guide lines with T square and triangles. 
No letters are to be put on the geometrical figures such as 
are shown in the following pages. They are placed there for 
descriptive purposes only. The student should print the 
number and title of each problem above the construction, as 
shown on the sample plate inserted in this Paper. All con- 
struction lines should be drawn lightly, so that they can be 
erased easily. Too much rubbing on a drawing not alone 
spoils the general appearance, but causes rough spots on the 
paper, on which dust will accumulate and adhere. 



DIRECTIONS FOR DRAWING PLATES I. TO V. 
INCLUSIVE. 

34. After you have carefully read the preceding articles 
and have practised the lettering and the use of the instru- 
ments, you begin to draw Plate I. A sample of this plate, 
on a reduced scale, is inserted in this Paper, and shows the 
general arrangement of the problems and the method of 
dividing the sheet of paper. This plate should be referred 



32 GEOMETRICAL DRAWING. 

to frequently, but the dimensions should be taken from the 
text, as the sample plate is not drawn full-size and gives no 
dimensions. In no case should a drawing be measured by 
student or workman; they should always be guided by the 
dimensions which are written on the drawing. For while 
the drawing itself may not be made exactly according to 
size, the dimensions are generally given in numbers which 
can be depended upon. 

Fasten your paper, which measures about 15* X 20", to the 
drawing-board, as explained in Article 16, and proceed 
to draw your cutting and working edges according to the 
sizes given in the last article. No part of the drawing 
proper should be placed between the cutting and working 
edges. 

Draw a light horizontal line midway between the two 
working edges, and then draw two vertical lines which will 
divide the space between the two vertical working edges into 
three equal spaces. This subdivision will give six spaces of 
6f" X about of" each, within which six problems are to 
be drawn. The lines are to be drawn very lightly, as they 
are not to be inked and must be erased. They are shown as 
dotted lines in the sample plate. Above each problem the 
number and title should be printed, as shown in the sample 
plate. The exact wording of the title is printed in the text 
above the description of each problem. The first line of 
lettering should begin ^ inch below the top line of each 
space. The capitals are -^ inch high, and the space between 
two lines of lettering is J inch. The highest point of the 
drawing should not be nearer than V inch below the lowest 
line of lettering, so that if there is one line of lettering only, 
the space between the top line of the space and the highest 
point of the drawing would be 1/ + ■£/ + V = I//- If 
there are two lines of lettering this space becomes -V -f -fa* 
_j_ ^." _|- ^" _j_ i/ = 1-Jig*. It is therefore necessary to 
determine by judgment whether the lettering will occupy 
one or two lines, before the problem can be drawn. The lat- 
ter should be placed centrally between the lowest line of 
lettering and the lower border line of the space. The letter- 



GEOMETRICAL DRAWING. 33 

ing should not be put in until all the problems on that plate 
have been drawn and inked in. 

Centrally between the upper cutting and working edges 
print the title of the drawing, which will be "Geometrical 
Drawing, Plate I.," for the first plate; " Geometrical Draw- 
ing, Plate II.," for the second, etc. These letters are all 
capitals, -f? inch high. In the right-hand corner, between 
the lower cutting and working edges, print your name, class 
letter and number, and in the left-hand corner the date. 
The sample page will show the manner of printing these. 

35. The student should make his drawing neatly and 
accurately. All dimensions should be closely followed and 
the lines be made exactly of the required lengths. If a line 
is to be drawn through two points, be sure that it does not 
pass alongside of either of them; and if two lines are to meet 
at a point, take care not to have them cross each other, but 
have the point sharp and decisive. In drawing a line tangent 
to an arc, do not let the line cut the arc, but have it touch only 
at one point. 

In the first five plates, which consist of construction prob- 
lems, a distinction has to be made between three kinds of 
lines, namely, the given lines, the construction lines, 
and the required lines. The given lines are made full 
and of ordinary width, that is, of a width shown in Fig. 25, 
line ab. The construction lines are dotted and of the same 
width as the given lines. The required lines are full and a 
trifle heavier than the given lines, as shown by line ce, 
Fig. 25. 

The student must ink in the curved lines before the straight 
lines, the dotted lines before the full ones, and the fine lines 
before the heavy ones. The working edges are made as' 
heavy as the required lines. 

After you have your pen adjusted for drawing lines of a 
certain width, draw all these lines before the pen is reset for 
another set of lines of a different width. Try your pen on 
the waste edges of your paper before you draw the lines. 

In actual working drawings the construction lines are 



34 GEOMETRICAL DRAWING. 

always erased, unless some difficult construction requires 
the lines to remain there for the guidance of the workman. 
In these problems, however, these lines are inked in dotted 
and should be made with great care, the dots being about ^ 
of an inch long. 

The division lines of the sheet and any unnecessary lines 
should be carefully erased before an}' inking is done, and if 
it is found necessary to use the rubber after the drawing has 
been inked, care should be taken not to erase any ink lines. 
If the latter does occur, you should go over that line with 
your pen, being sure to make it the required width. 

In using your compasses be sure you have a sharp needle- 
point. Bear lightly on your compasses or dividers, so as not 
to make large holes in the paper. 

Again, do your lettering with great care and observe the 
construction of each letter. Do not hurry your work, but be 
satisfied with slow advancement at the beginning, and you 
will find that the careful worker will reap the greatest bene- 
fits from his studies and will have no difficulty in drawing 
the plates in the latter part of this course. 

After you have completed your drawing, cut it off along 
the cutting edges, but do not use your T square as a guiding 
edge for your knife. Cut accurately along the lines, and use 
a sharp knife or a pair of scissors. 

Write your name and address on the back of the plate 
with pencil, put it into one of the mailing tubes which we 
sent you, and mail it to us. 

Then begin your work on Plate II.. observing the above 
rules, but do not send this to us until you have received 
Plate I. back from us and have obtained a passing mark. 
Read and profit by the suggestions made by the instructor 
on your first plate while drawing Plate II. 



GE03IETRICAL PROBLE3IS. PLATE I. 

(The student should refer to the sample plate.) 

36. Problem 1. To bisect (bi-sect) a straight line 

or an ore. 



GEOMETRICAL DRAWING. 



35 



* 



M 



Fig. 24. 



Let ab, Fig. 24, be the given 

line or arc, the line being 4" long. 

From a as a center, with a ra- 
dius greater than half of ab, 

describe arcs c and d. From b 

as center, with the same radius, 

describe arcs cutting the former 

in c and d. Through these 

points of intersection draw the 

line cd, which will divide the 

line and arc ab into two equal 

parts. 
Problem 2. To draw a perpendicular to a straight 

line when the point is at or near the end of the line, 

•Let ab, Fig. 25, be the given 
line, which is to be drawn 4" 
long. Let c be the point at, 
I which the perpendicular is to be 
I drawn, and let it be \" from b. 
Take any point o above the line 
ab as a center, and with the dis- 
tance oc as a radius describe the 
^~~ — --"'' arc ecd cutting ab in d and c. 

Draw the line od and produce it 

until it cuts the arc at e. Join e and c by a line, and ec will 

be the perpendicular required. 

Problem 3. To draw a perpendicular to a straight 

line from a point without it. 
Let ab, Fig! 26, be the given 

line, 4" long, and d the given 

point. From any point c in the 

line ab, about f" from b, as a 

center, and with a radius cd, 

describe the arc de cutting ab in 

/. With / as a center and the 

radius fd, describe arcs cutting 

the first arc at d and e. Through 

the points d and e draw the line 

de, which will be the'perpendicu- 

lar required. 



~dX 



36 



GEOMETRICAL DRAWING. 



Problem 4. To draw a straight line parallel to a 

given straight line. 

g ;-- i , v - -. h Let ab, Fig. 27, be the given 

ine, 4 " long. From any two 
points on that line, as c and cL as 
centers, and with a radius equal 
to the given distance between'the 
two lines, which is to be about 

1J ", describe the arcs e and /. Draw the line gh tangent to 

the arcs e and /. The straight line gh is parallel to ab and 

a given distance from it. 





Problem 5. To bisect a given angle. 

Let aob, Fig. 28, be the given 
angle. With the vertex o as a 
center, and any radius, describe an 
arc cutting the sides of the angle 
at c and d. From c and d as cen- 
ters, with the same or any radius, 
describe arcs cutting each other 
at e. Through this point of inter- 
section draw the line oe, which 
bisects the given angle. 

Problem 6. To bisect the inclination (in-cli-nation) 
of two straight lines, the vertex (vertex) of which is 
inaccessible (in-ac-ces'si-ble). 

Let ab and cd, Fig. 29, be the 
_-/ given lines, 3V long. Draw ef 
., and gh, respectively, parallel to 
ab and cd, by the method given 
^ /l in Problem 4, ef being the same 
-a distance from ab as gh is from 
FlG M - cd, and intersecting each other at 

o. Bisect the angle fuh by the method given in Problem"), 
and the line oi, which bisects this angle, also bisects the in- 
clination between the given lines. 



GEOMETRICAL DRAWING. 37 



GEOMETRICAL PROBLEMS. PLATE II. 

37. The cutting, working, and division lines are laid out 
the same as for Plate I. , and the student should carefully 
reread the instructions contained in Articles 34 and 35 and 
profit by the criticisms the instructor has made on Plate I. 
Do not send this plate to the Schools until Plate I. has been 
returned to you. 

Problem 7. To divide a straight line into a required 
number of equal parts. Let ab, c 

Fig. 30, be the given straight „ £ ''' 

line, 31" long, to be divided into ^ fk^\ \ 

eight equal parts. From a draw */ { \ \ \ 

an indefinite straight line, ac, 2 ^' \ \ \ \ \ 

forming an angle with ab. Set ^ '' \ \ \ \ \ \ \ 
off on the line ac eight equal a i' & 3' # s r «' r b 
parts of any length. Join the FlG " 80 ' 

points 8 and b by a straight line, and draw lines parallel to it 
through the points 1. 2, 3, 4, 5, 6, 7, and they will divide ab 
into the required number of parts, a 1', 1' 2', etc., all being 
equal to each other, and each equal to one-eighth of the 
distance ab. 

Problem 8. To draw a straight line to form any re- 
quired angle with another straight line from a given 
point in it. 

Let cd, Fig. 31, be the given 

^ line, 4" long, e the given point, 

^^ and aob the given angle. 

J\ From o as center, with any 

^/^ \ convenient radius, describe an 

If arc ba. From e as center, and 

FlG ' 81 " with the same radius, describe 

the arc gf. From / as center, with a radius equal to the 

chord of the arc ba, describe an arc intersecting gf in g. 

Through the points g and e draw the straight line eg, which 

is the required line to make the angle gef = boa. 



38 



GEOMETRICAL DRAWING. 





Problem 9. To construct 
an equilateral triangle, one 
side being given. 

Let ab, Fig. 32 s be the given 
side, 3$" long. From a and b as 
centers, with a radius equal to 
ab, describe arcs cutting each 
other in c. Draw the lines ac 
and be. The triangle abc is 
FlG - 32 - equilateral, that is, ab = ac =* be. 

Problem 10. To construct an equilateral triangle, 
the altitude being given. 

Let cd, Fig. 33, be the given 
altitude, equal to 3". Through 
the point d draw a straight line, 
ef, perpendicular to cd, by the 
method shown in Problem 2. 
Through the point c draw another 
straight line, ab, parallel to ef, by 
the method shown in Problem -i. FlG - 33 - 

From das a center, and any convenient radius, describe a 
semicircle cutting ef in e and /. From e and / as centers, 
with the same radius, intersect the semicircle in g and/?. 
From d and through the points g and h draw the lines dg 
and dh, and extend them until they meet the line ab. 

Problem 11. To construct a triangle, two sides and 
the included angle being given. 

Let ab and cd, Fig. 34, be the 
two given sides, 3-i -'" and 3" long 
respectively, and e the given an- 
gle, about 60°. 

Draw the line fg equal to ab. 
At the point / construct an angle 
equal to e, by the method given 
in Problem 8, and make fh equal 
to cd. Join the points h and </, 
1 l<; !l and fgh is the triangle required. 

Problem 12. To construct a parallelogram, the 
length of the sides and one of the angles being given. 




GEOMETRICAL DRAWING. 



39 



Let ab and cd, Fig. 35, be the lengths of the two sides, 





4" and 3-J-" respectively, and e the 

given angle, about 60°. Draw gf 

equal in length to ab. From / 

draw fh, equal in length to cd, 

and forming with gf an angle 

equal to the given angle e. From 

the point g as a center, with a fig. 35. 

radius equal to cd, and from h with a radius equal to ab, 

describe arcs intersecting at i. Draw hi and gi, and gfhi 

is the parallelogram required. 

Note. — Do not mail this plate to us until Plate I. has 
been returned to you. 

GEOMETRICAL PROBLEMS. PLATE III. 

38. The directions for dividing the sheet of paper given 
for Plates I. and II. also apply to this plate, as well as to 
Plates IV. and V. 

Problem 13. Given an arc of 
a circle, to find the center. 

Let ab, Fig. 36, be the given arc. 
Take any three points on the arc, 
as a, c, b. Bisect the distances ac 
and cb by the lines fg and ed, and 
the intersection of these lines at o 
}fc will be the center of the circle of 

fig. 36. which the given arc is a part. 

Problem 14. In a given 
circle to inscribe a square. 

Let abed, Fig. 37, be the given 
circle, 4" in diameter. Draw two 
diameters, ac and bd, at* right 
angles to each other, by the 
method shown in Problem 1. 
Join the points a, b, c, d, where 
the diameters intersect the cir- 
cumference, and these lines will 
be the sides of the square. 





40 



GEOMETRICAL DRAWING. 



Problem 15. In a given circle to inscribe a regular 
j hexagon (hex'a-gon). 

In Fig. 38, let o be the center 
of the given circle, 4" in dia- 
meter. Draw the diameter ab. 
_•___ ___ ^|6 From the points a and b as cen- 

ters, and a radius equal to ao or 
ob, the radius of the given circle, 
describe arcs cutting the circum- 
ference of the circle in the points 
c, d, e, f. Join the points a, c, 
f, b, e, and d, and acfbed will be the required hexagon. 

Problem 16. In a given circle to inscribe a regular 
pentagon (pent'a-gon). 

Let abed, Fig. 39, be the given 
circle, 4" in diameter. Draw the 
diameters ab and cd at right 
angles to each other, and bisect 
ob in the point e. From the < 
point e as center, and with a ra- 
dius equal to ec, describe an arc 
cutting ab in/. From the point c 
as center, and with a radius cf, de- 
scribe an arc cutting the circum- 
ference of the circle in g. Join FlG - 39 - 
the points c and g. Then draw the chords eli. In'. ij,jg, 
each equal to gc, and gchij is the pentagon required. 
Problem 17. In a given circle to inscribe a regular 
heptagon (hepta-gon). 

Let o, Fig. 40, be the centre of 
the given circle, which is 4" in 
diameter. From any point on 
the circumference of the circle, 
as a, as a center, and with a radius 
/ ao, equal to the radius of the circle, 
describe an arc, cod, cutting the 
circumference in c and d. Draw 
the chord cd. and join ao, which 
bisects cd in the point e. Set off 
from a a distance equal to ce or 





GEOMETRICAL DRAWING. 



41 



de, around the circle, and by joining the points the heptagon 
akjihgf will be completed. 

Problem 18. In a given circle to inscribe a regular 
polygon (pol'y-gon) of any num- 
ber of sides. 

In Fig. 41 describe a circle 
with a diameter of 3|", and draw 
the diameter &9. Divide this 
into as many equal parts as the 
proposed polygon has sides (in 
this case nine). Bisect 69 in o, 
and draw o3" perpendicular to 
b9, making 4/3", the part with- 
out the circle, equal to three- 
fourths of the radius bo or o4 . 
the 
the 
second division from b on the 
diameter &9, producing it to meet the circumference in a. 
Join a and b, and ab will be one side of the required poly- 
gon. The others may be stepped off around the circum- 
ference. 



From the point 3 " draw 
straight line 3 "a through 




GEOMETRICAL PROBLEMS. PLATE IV. 

39. The preliminary directions for this plate are the 
same as for the preceding ones, and the student should pro- 
fit by the criticisms made by the instructors on those plates. 



Problem 19. Given one side of a regular polygon, 
to construct the polygon. 

Let ab, Fig. 42, be the given side, 3|" long. Produce ab 
to c, making be equal to ab. From b as a center, with a 
radius ab, describe a semicircle. Divide this into as many 
equal parts as there are sides in the proposed polygon (in this 
case five). From the point b, and through the second divi- 
sion from c, draw the straight line 63. Bisect the lines ab 
and 63 by perpendiculars intersecting in o. From o as a 



GEOMETRICAL DRAWING. 




center, with a radius oa, 
ob, or o3, describe a 
circle. From b, and 
through the remaining 
divisions in the semi- 
circle a3c, draw lines 
till they meet the cir- 
cumference in / and g. 
Join the points 3g, gf, 
and fa, and ab'dgf is the 
polygon required. 
To dratv a tangent to an arc or circle 



Problem 20. 

at a given point. 

Let acb, Fig. 43, be the arc, « 
and c the given point. From 
c to the center o draw the radius 
co. Through c, perpendicular 
to co, draw the line cd, which 
is the required tangent. 

Problem 21. To draw a line tangent to two arcs, 
and to find the points of tangency (tan'gen-cy). 

Let ab and cd, Fig. 44, 
be the given arcs. Draw 
the line ef tangent to both 
arcs. From the centers o' 
ji and o draw perpendiculars 
to ef. The points e and /, 
where these lines meet the 
line ef, are the points of tangency between ef and the two arcs. 




Problem 22. To find the point 
of contact of tivo tangent arcs or 
circles. 

Let ab and cd, Fig. 45, be the two 
arcs tangent to each other. Join their 
centers o and o'; and the point e, where 
the line od cuts the arcs, is the point of 
contact required. 



GEOMETRICAL DRAWING. 



43 



Problem 23. To draw a straight line equal in length 
to any given arc. 

Let ab, Fig. 46, be the given . 

arc. Draw a straight line con- 
necting the center of the arc, o, 
with its one extremity, a. Draw 
ac perpendicular to oa, and divide 
the arc into four equal . parts. 
With a as a center, and a radius 
equal to the chord of a\, describe 
an arc cutting ac in d. With d 
as a center, and db as a radius, de- 




scribe an arc cutting ac in c. Then ac is the required length. 

Problem 24. To find the arc of a circle ivith a given ra- 
dius, which shall be equal in length to a given straight line. 
2 3b Let ab, Fig. 47, be the given 
line. At its one extremity, a, 
erect a perpendicular, ao, and 
make it equal in length to the 
given radius. With o as a center, 
and a radius equal to ao, describe 
an arc. Divide the straight line 
ab into four equal parts, and with 
the first division, marked 1, as a 
center, and 16 as a radius, de- 
scribe an arc cutting ac in c. Then ac is nearly equal in 
length to ab. 

Note : Problems 23 and 24 are only approximately correct, and should 
only be depended upon when the arc is less than about one-sixth of 
the circumference of the circle. For larger arcs numerical calcula- 
tions would give the most satisfactory results. 




GEOMETRICAL PROBLEMS. PLATE V. 

40. There are four problems on this plate, instead of 
six, the last two problems requiring two vertical spaces each 
for their construction. Therefore the student should divide 
the sheet of paper into three equal parts vertically, and the 
first of these should be divided into two equal spaces by a 
horizontal line. 



44 



GE< 'METRICAL DRAWING. 




Problem 25. — To describe an ellipse (el-lipse), the 
two diameters being given. 

In Fig. 4^. draw the two given diameters, ab 3V long and 
A cd 2f" long, at right angles to 

each other, intersecting at their 
point of bisection o. From o as a 
center, with oa or ab as radius, 
describe a circle : and from the 
( same point as center, with oc or 
od as radius, describe another cir- 
cle. Divide both circles into the 
same number of equal parts. 1.2. 
3. 4. etc.. and 1. 2.3. c, etc. 
* 4 ~> This is best done by dividing the 

upper half of the larger circle into 
the required number of parts, and then drawing radial lines 
from these divisions through the center, dividing the lower 
half of the larger circle and the entire smaller circle into the 
same number of parts. From the points of division of the 
larger circle draw lines perpendicular to ab, as le. '2f, 3g. 
etc . and from the corresponding divisions of the smaller cir- 
cle draw lines parallel to ab, as el . f% . go . etc.. cutting the 
perpendicular lines in the points e, f, q, etc. Through the 
points of intersection of these lines draw the circumference 
of the required ellipse. It is advisable for the student to 
first sketch the ellipse, through the points found, with a pen- 
cil before he applies his irregular curve. This latter should 
be used as was explained in Art. 21. 

Problem 26. Given the diameters, to describe an 
ellipse by circular ■■ 

Let ab and cd. Fig. 49, be the 
given diameters., ab being 34* 
long and cd $J . at right angles 
to and bisecting each other at o. 

From c on cd, at a distance 

than CO, lay off a distance 

From a on ab lay off the 

distance ae = cf. Join e and 

/ by a straight line. Bisect the 

line ef by hg, which will be perpendicular to ef and cut cd 




GEOMETRICAL DRAWING. 



45 



in g. 



Through the points g and e draw a straight line geh". 
, og' = og and draw the line geh". Also make oe' = 
oe, and draw the lines gh' and g'h . From g and g' as cen- 
ters, and a radius equal to gc or g'd, describe the arcs h"ch 
and h"dh', respectively. 
From e and e as cen- 
ters, and ae or e7> as 
radius, describe the arcs 
h'ah" and h'bh', respec- 
tively. Then acbd is 
the required ellipse. 

Problem 27. Giv- n 
en the diameter and io- 
pitch, to draw a helix 9 
(he'lix). 

Definition. The 
helix is a curve form- 
ed by a point travel- 
ing around a cylinder^ 
and, while thus moving, 
advancing a certain uni- 
form distance along the 
length of the cylinder. 
The winding curve 
shown in Fig. 50 is thus 
produced, making one 
complete revolution 
around the cylinder in 
the space cl2, which is 
called the pitch of the 12] 
helix. The line a9' is 
called the axis, and is 
also the axis of the cyl- 
inder around which the 
helix is described. The 
perpendicular distances 

from the axis to any fig. m. 

point of the helix are all equal to the radius of the cylinder. 

Construction of the helix. Let cdef, Fig. 50, rep- 




46 GEOMETRICAL DRAWING. 

resent the side view of a cylinder on which the helix is sit- 
uated, and let its diameter be 4"; ao is the axis and cl2 is the 
pitch = 2£*. One and a half turns of the helix are to be 
drawn, making the length of the cylinder, ce or df, = 3f ". 
Draw the line ef 4' long, 3 " from the top border line, and 
make ec and fd perpendicular to it and 3£" long. After you 
have completed the side view of the cylinder, namely, the 
rectangle cefd, you draw the bottom view of the cylinder, 
which is the circle below, with the center o. This center is 
located 2^ " below the line cd, and the circle is 4' in diameter. 

Draw the axis ao and the diameter 12 '06'. Divide the 
circle into any number of equal parts (12 in this case) begin- 
ning at 12', the one extremity of the diameter 12 06'. We 
may remark that the greater the number of these divisions, 
the greater will be the accuracy of the curve. 

Set off, on ce, the pitch of the helix, namely, cl2, equal to 
2| ", and divide it into the same number of equal parts as the 
circle (12 in this case), marked 1, 2, 3, 4, 5, 6, etc., in the 
figure. 

Then through the points of division of the circle draw 
straight fines perpendicular to cd or parallel to the axis ao, 
and through the points of division of the pitch draw straight 
lines parallel to cd or perpendicular to the axis ao. The 
points of intersection of the corresponding pairs of these two 
sets of lines will be points in the required curve, as 1'. 2 . 3 . 
4 . 5 . etc., through which the curve should be drawn with 
an irregular curve. The part of the helix from 6" to 12 is 
dotted because it is located on the back half of the cylinder 
and cannot be seen by looking at the front of the cylinder. 
Anything which is hidden is represented by dotted lines in 
mechanical drawings. 

Problem 28. — To draiv the Ionic (i-on'ic) volute 
(vo-lute'). the height being given. 

This problem occupies the last two vertical spaces on this 
plate, and consists of the upper figure, which is the volute 
proper, and the lower figure, which show r s the enlarged con- 
struction within the eye of the volute, which is the circular 
termination at the center. 



GEOMETRICAL DRAWING. 



4? 



Divide the space on your paper into two equal parts by 
drawing the vertical line cle (Fig. 51), which will also cut the 
lower figure in a and d. 2" to the right of this, and 4" below 



rejjej % / f fi . sr f y <f c 




the top border line, draw the 

vertical line ab, which is the height 

of the volute and is 4" long. 

Divide this line into seven equal 

parts, asal', 1' 2' , etc., and through 

the third division, from the bottom, 

draw the horizontal line 3 c. From 

the point of intersection of this 

line and the line ed as a center, 

and with a radius equal to one-half 

of one of the equal divisions on ab, fig. sia. 

describe a circle, which forms the eye of the volute. 

For finding the centers for the twelve quadrants of the volute 
a constructionis employed which is shown enlarged in Fig. 51a. 




4s GEOMETRICAL DRAWING. 

On the line ad, which is a prolongation of the line de, 
and 3f " below the center of the eye of the volute, we take 
another center and describe a circle having a diameter of 
2". Within this we make the following construction: 

Draw the horizontal diameter cb perpendicular to ad at 
its middle point, and join c, a, b, and d by straight lines. 
From 1, the middle point of aj), to 2, the middle point of ac, 
draw the horizontal line 1, 2. Divide the distance between 
the lines cb and 1. 2 into 3 equal parts, and lay off 2-h of these 
parts below the line cb, and draw the line 3, 4, the points 2 
and 3 being joined by a perpendicular line. Then, with your 
45° triangle, draw lines from 1 and 2 toward the center, and 
also from 3, which meets the line cb -% space to the left of the 
center. From this point of intersection draw another line at 
45°, meeting the line 3, 4 at 4. Then draw the vertical and 
horizontal lines as shown, all terminating at the 45 c lines. 
The points thus obtained are marked 1. 2, 3, etc., up to 12. 
These will be twelve centers for the twelve quadrants, which 
will give three complete turns to the volute. 

After the student thoroughly understands this construc- 
tion, he should insert a similar one into the eye of the volute. 
Then draw the horizontal line bf (Fig. 51), and, with 1 as a 
center and a radius equal to If, describe the quadrant fg. 
With 2 as center, and a radius 2g, draw the quadrant gh. 
With 3 as center, and a radius 3g, draw the quadrant hi. 
With 4 as a center, and a radius 4t, draw the quadrant ij, 
and continue this operation with every center, the last being 
12, when a radius of 12q describes the quadrant from q until 
it becomes tangent to the eye. 

This completes the drawing of the outer volute, and it now 
remains to draw the inner one. For this purpose lay off bb 
= f". From b draw a line meeting bf in any point, such 
as ri , a convenient distance being IV. Divide bri or b'n 
into 12 equal parts and join the corresponding divisions by 
vertical lines as shown, giving the distances bb' , c'c', d'd', 
e'e'yff, etc., up to mm'. Then draw the horizontal line b'r 
intersecting the line /l in r, the distance //• being equal to 
bb'. Lay off a distance equal to c'c' on the line g2, giving 
the point s. Lay off d'd' on the line Sh, giving the point t; 



GEOMETRICAL DRAWING. 49 

lay off e'e on M, giving the point u; lay off /'/' on 5j, giving 
the point v, and continue this until mm' is laid off on 12q, 
giving q 1 . In order, then, that the last quadrant will be 
tangent to the eye, and the first quadrant being a distance 
equal to bb' from the former volute, the centers must neces- 
sarily be shifted. This is accomplished as follows: The point 
r has been obtained. The first quadrant has to pass through 
the two points r and s Therefore, with a radius equal to 
lr, the compasses are moved along 1, 2 until the new lr is 
equal to the new Is, which center will be found to be a trifle 
to the left of the first center marked 1. The next quadrant 
has to pass through the points s and t. Hence the center 
will be found a trifle below the old center 2 on the line 2h, 
the radius being 2s. The next center is a trifle to the right 
of 3 on the line 3i, and the radius is St. This operation is 
continued until the inner volute is completed. The student 
should be sure to have the quadrants meet each other with- 
out forming ugly corners or overlapping each other; the 
construction should be made with great care. 

PRACTICAL HINTS ON THE USE OF 
INSTRUMENTS. 

4 1 . The instruments and their principal uses have been 
fully discussed in the opening chapter of this Paper, and by 
a correct application of the rules laid down therein, and a 
careful study of the geometrical problems just drawn, the 
student should be able to make drawings as neatly and 
accurately as an experienced draughtsman. However, there 
is no profession which does not pride itself on the number of 
" kinks " or "practical hints" which have been acquired by 
years of experience and practice. And so the draughtsman 
and architect have discovered many "short cuts "for the 
execution of some particular problem, or some novel and 
helpful ways of using the instruments at their disposal. 

We would not be fulfilling our mission if we should omit 
reference to some of these thoroughly practical uses of the 
instruments, and while the student will acquire a number of 



50 



GEOMETRICAL DRAWING. 



them himself hy close observation and his own ingenuity, 
the following hints may be of service to him in his work. 

42. The student must have observed, while constructing 
the geometrical problems,, that the only instruments used, 
with but few exceptions, were the T square and compasses. 
The scale was, of course, used for measuring the lengths of 
the given lines, and the triangle for the drawing of vertical 
lines in the last few problems: but it must have occurred to 
all, that most of the methods described could be used when 
nothing is at hand but a straight-edge, a pair of compasses, 
and a pencil. However, compasses are not always at one's 
disposal when the T square, triangles, scale, and pencil are. 
The following constructions will be helpful in these cases, 
and are simpler and therefore more rapid than the construc- 
tions just drawn. However, we wish to impress this fact 
firmly on the mind of each student: that the principles under- 
lying the construction of the geometrical problems are very 
essential to the education of every craftsman, as all con- 
structions, of whatever nature, are based on geometrical 
principles. 

43. To divide the distance between two lines into 
any number of equal parts. 




Let ab and cd, Fig. 52. be the given lines, and let it be 
required to divide the distance between them into five equal 



GEOMETRICAL DRAWING. 



51 



parts. Lay you* scale, A, diagonally across the distance, as 
shown in the figure, so that the diagonal distance embraces 
an equal number of divisions, as five inches or five one-half 
inches = 2J- ff . Then mark off the points e,f, g, h on the 
even inch marks, a' and b' being on the lines ab and cd. 
Through the points e, f, g, h draw lines parallel to the given 
lines, and the distance between them will be divided into 
five equal parts. 

44. To divide a circle into a given number of equal 
parts by the use of triangles. 

For inscribing or circumscribing (cir-cum-scri'bing) regu- 
lar polygons having e 
4, 6, 8, or 12 sides, 
the 45° and 60° tri- 
angles may be em- 
ployed as shown in 
Fig. 53. 

For convenience 
of demonstration 
the circle has been 
divided into quad- 
rants. The one 
marked A shows 
how the side of an 
inscribed or circum- 
scribed square may 
be drawn by using 

the 45° triangle. riu -°° 

The inscribed side is limited by the extremities c and b of 
the diameters; the side of the circumscribed square is tan- 
gent to the circle and is limited by the prolongation of the 
diameters ab and cd at e and /. 

Quadrant B is bisected by means of the 45° triangle, and 
in this way two sides of an octagon, eg and ga, are obtained. 

By using the 60° triangle in two different positions we 
divide quadrant C into three equal parts, or the entire 
circle into twelve parts. 

In quadrant D the 60° triangle is used in one position 




52 



GEOMETRICAL DRAWING. 



only, and we obtain dj, one side of a hexagen, and jk, one- 
half of another side. This enables us to inscribe or circum- 
scribe a hexagon. 

The student has, therefore, three methods at his disposal 
for dividing circles — namely, by using (1) the protractor, 
(2) the geometrical problems, (3) the triangles. 



45. Further use of the triangles. 

As has been pointed out in a previous chapter, triangles 
are used for drawing lines which have an inclination of 30 c , 
45°, 60°, and 90° to other lines, and for drawing parallels 
and perpendiculars to lines having any inclination what- 
ever, as was pointed out in Arts. 5 and 6. However, 
other angles besides those mentioned above may be obtained 
by means of the 45° and 60° triangles, a knowledge of which 
may be of service when no protractor is at hand. 
r — _ If it is desired to obtain an angle 

of 15°, place your 45° triangle in 
the position aoc, Fig. 54, and draw 
oa and oc. Then place your 60° 
triangle in th position hoc, the 
vertex of the 30° angle coinciding 
with the vertex of 45° angle just 
drawn, and one side being coinci- 
dent with the side oc of the 45 
angle. Draw ob, and the angle 
= 15°. 




Fig. 54. 

aob will be equal to 45° - 30° 



For drawing an angle of 75° the 
triangles are placed in the position 
shown in Fig. 55. The angle cob 
is equal to 45°; the angle boa is 
equal to 30°, and the total inclina- 
tion between oc and oa is equal to 
45° + 30° = 75°. 




For drawing an angle of 1(>5° the triangles A and />\ 45 




GEOMETRICAL DRAWING. 53 

and 60° respectively, are placed in the positions shown in Fig. 

56, which explains itself, the total 

angle being 60° + 45° = 105°. 

For reference, we give the fol- 
lowing table of angles, which can be 

easily obtained by using the triangles 

as shown in the above examples. 

Angles of 30°, 45°, 60°, 90° can be 

obtained directly, from the use of 

the 45° and 60° triangles. Angles of 

15°, 75°, 105°, etc., are obtained by 

using the following combinations : FIG ' 56 ' 

45° - 30° = 15° . 90° + 45° = 135° 

45° + 30° = 75° 90° + 60° = 150° 

60° + 45° = 105° 90° + 45° + 30° = 165° 

90° + 30° = 120° Straight line = 180° 

46. Methods for dividing a circle into any number 
of equal parts by using protractor or dividers. 

Whenever a circle is to be divided into any number of parts 
which are factors of 360°, a protractor can be used in the 
manner explained in Art. 22. For example, to divide a cir- 
cle into five equal parts, make each part equal to- 3 -!- - = 72°. 

However, for dividing a circle into five parts the geometri- 
cal method should be used. For dividing a circle into three 
parts or multiples of three, the 60° triangle is the most con- 
venient to use; for seven parts the geometrical method is 
used, and for other divisions up to thirteen, which is a safe 
limit, Problem 18 should be made use of. For dividing a 
circle, for example, into fifteen parts, it is advisable to first 
divide it into five parts and then to trisect one of these 
parts. For obtaining fourteen parts or any other multiple 
of seven, as forty-nine, divide the circle into seven parts 
and then subdivide one of these parts with the dividers. 
This method is more rapid than dividing the entire circle 
into the required number of parts, and the liability of error 
is decreased to a minimum. 

The student should use his own judgment after these few 
suggestions as to when and how to use the protractor, the 
geometrical problems, the triangles, and the dividers for di- 
viding a circle into any number of equal parts, and after a care- 



54 



GEOMETRICAL DRAWING. 



ful study of the various methods he will be enabled to choose 
the one which is the most expedient in any particular case. 

PRINCIPLES OF PROJECTION. 

47. Let us take a piece of wire, ab, Fig. 57, and hold it 
up vertically some distance away from a plane surface, as 
shown in the figure. If we now 
look down on the wire in the direc- 
tion indicated by the arrow, we 
see nothing but a point. If the 
wire was of an indefinite length it 
would pierce the plane defg in the 
point c. This point is called the 
projection of the line ab on the 
plane defg. In other words, the 
projection of a point on a plane is 
the point where a perpendicular 

from the point to the plane pierces „ f ?> 

that plane. If we now place the 
wire into the position ab, Fig. 58, 
all that is necessary to project that 
line on the plane is to project its 
two extremities, a and b, as ex- 
plained above. The perpendicu- FlG - 5S - 
lars aa' and bb pierce the plane in a' and b , and joining 
these by the line ab' we get the projection of the line ab on 
defg. And it matters not what the position or shape of the 
line may be, a line joining the perpendicular projections on 
the plane will be the projection of the line on that plane. 

In Fig. 50, for example, we have 
bent the wire ab into the circle 
acb, whose axis is perpendicular 
to the plane defg. Projecting 
several points, as explained above, 
we find that the perpendiculars 
pierce the plane in points which, 
if joined by a line, will form the 
circle acb. So much for the 





projection of points and lines. 



GEOMETRICAL DRAWING. 





Now let it be required to project a surface upon a plane. 
Let a surface stand in an upright 
position, as shown in Fig. 60. If 
we look down upon it, as indicated 
by the arrow, we simply see a 
straight line ab, and if the surface 
were prolonged indefinitely it , 
would pierce the plane efgh in the 
straight line ab'. This is called 
the projection of the surface abed 

upon the plane efgh. It is obtained by projecting the line 
ab upon the plane efgh, as previously explained. 

If we now place the surface in the 
position shown in Fig. 61 and pro- 
ject the lines which bound it,"we 
find the projection of abed on efgh 
to be a'b'c'd'. The projection of 
ab is a'b'; of be, b'c'; of cd, c'd'; 
and of ad, a'd'. 
FlG - 61 - Hence we obtain the rule: To 

project a surface upon a plane, project the lines which 
bound the surface. It matters not what position the sur- 
face occupies or of what shape it is, this rule will always 
apply. 

To show this, we refer the stu- 
dent to Fig. 62, where we have to 
project the curved surface abed 
upon the plane efgh. Again pro- 
jecting the lines ab, bd, cd, and 
ac, we obtain the figure a'b'c'd', 
which is the required projection. 
If it be required to project a point fig. 62. 

m in the plane abed upon the plane 

efgh, the same rules that have already been given apply. 
That is, a perpendicular is dropped from the point to the 
plane, and where that perpendicular pierces that plane is 
the projection of that point, which in our case is m'. 

Continuing this reasoning, the student will see that the 
projection of a solid on a plane is the projection of the 




56 



GEOMETRICAL DRAWING. 



surfaces that bound that solid. Hence, to sum up all the 
principles, we may say: 

(1) The projection of a point upon a plane is to find where 
a perpendicular from the point to the plane pierces the plane. 

(2) The projection of a line upon a plane is the projection 
of the points, which compose that line, upon the plane. 

(3) The projection of a surface upon a plane is the projec- 
tion of the lines, which bound the surface, upon the plane. 

(4) The projection of a solid upon a plane is the projection 
of the surfaces, which bound the solid, upon the plane. 



48. A mechanical drawing, intended to guide the 
mechanic in the construction of objects, should be so made 
that nothing is left to his imagination and that he can obtain 
the exact size and shape of each detail from the various 
dimensions and views. This necessitates each line to be 
represented in its true length on some part of the drawing, 
and a sufficient number of views of the object to show its 
true outlines. That this cannot be accomplished by a per- 
spective drawing can readily be seen by reference to Fig. 63. 
While a person can get a good general idea, from such a 
drawing, as to what it is meant to represent, a mechanic 
could not work from it, on account of 
the fore-shortening of some of the sides. 
Besides, perspective drawings of com- 
plicated objects are very difficult to 
make, and become so complicated as to 
be unintelligible to a workingman. 

We therefore resort to projection, 
which enables us to overcome the diffi- 
culties met with in perspective draw- 
FlG - 63 - ings. By applying the principles of 

projection presented in the last article we are enabled to 
make a sufficient number of views of an object to give a 
person a correct understanding of its shape and size. 

The various views of an object are named according to the 
direction from which the draughtsman was supposed to be 
looking at the object while drawing that view. The more 
complex the shape of an object, the more views are required 




GEOMETRICAL DRAWING. 



57 



to represent it on a drawing. We may either look at the 
top of an object and get the top view or plan. Looking 
at the side of an object we get the side elevation 
(el-e-va'tion) or side view, and looking at the end the 
end view or end elevation is obtained. Sometimes it 
is advisable, in order to show some detail of construction, to 
make a section of an object, that is, to image the object cut 
by a plane along a certain line, and then make a drawing of 
the part by looking toward the surface which has been cut. 
According to whether the cut is made along the length or 
through the width or height of the object, we obtain longi- 
tudinal (lon-gi-tu'di-nal) and vertical or transverse 
{trans-verse') sections. 

49. Generally two or three views are sufficient to show 

a 

A I 




-W 






S' 



the shape or size of 
an object, namely, the 
Plan, the Side Eleva- 
tion, and the End Ele- 
vation. For circular or 
symmetrical figures 
the latter may be omit- 
ted. 

fig. 64. To show the arrange- 

ment of these views, 
let us take the object shown in perspective in Fig. 63. 

The three views shown in Fig. 64 are the plan, in the upper 



58 GEOMETRICAL DRAWING. 

left-hand corner, the front elevation below it, and the side 
elevation to the right of it. 

Before the student begins a drawing he has to determine 
the following: the number of views necessary, how to place 
the views, and how to space them on the sheet of paper. 
The various views are connected by center lines, as ab and cd 
in the figure, which, as their names indicate, pass through 
the center of each view and through the center of every 
circle, where they intersect each other at right angles. These 
lines are broken lines and consist of dashes and dots. 

To draw the object shown in Fig. 64, it is found necessary 
to draw the plan first, on account of the hexagonal shape of 
the base. After you have, therefore, decided where to place 
the center o, draw the center lines ab and &d perpendicular 
to each other. Around this center o construct the hexagon 
ABCDEF, and describe the two circles representing the out- 
side and inside circumferences of the cylindrical extension 
KLHN (shown in the front elevation). This completes the 
plan of the object, and the next view to draw is either the 
front or side elevation. Let us draw the front elevation 
first, wh'ich is located below the plan. 

First draw the base line F'C of indefinite length, and pro- 
ject the points F and C on this line by drawing the vertical 
lines FF' and CC. Lay off F'F' and CC equal to the 
thickness of the base and draw the upper line F'C. Project 
the points E and D on the lines F' C and draw the lines E' E 
and D D . Then draw the line K'L' at the required distance 
from F'C : project the points iTand L on this line, and obtain 
the points K and L . Draw K'H and L' X parallel to the cen- 
ter line ab. Project the smaller circle — that is, the hole 
through the cylinder and base — in a similar manner by pro- 
jecting the extremities of the horizontal diameter, and draw 
the dotted lines BS and PT parallel to the center lino ab. 
These lines are dotted because they cannot be seen by looking 
at the object in the direction which gives the front elevation. 

To draw the side elevation we proceed as we did for the 
front elevation, by first drawing the base line D'B', then 
drawing the upper line D'B' the required distance from it . and 
joining them by the linos /IB' and D' ' D" ; the hollow cylinder 



GEOMETRICAL DRAWING. 59 

is projected as it was in the front elevation. The lines B D" 
will be shorter than the lines F'C in the front elevation, 
because the distance between the sides of a hexagon is less 
than the distance between the corners. The absence of the 
lines E'E' and D'D' and the presence of the line C" C" in the 
side elevation, also needs no special mention. Reference to 
the figure will explain this difference between the two eleva- 
tions. 

After studying the principles of projection and the arrange- 
ment of views, as illustrated by the object just drawn, the 
student should be in a position to represent any object on 
paper by as many views as are necessary for determining its 
exact shape and size. He can now begin, after studying the 
next chapter, on the next series of plates, which consists of 
a plate on projection, one on projections and conic 
sections, each problem representing some new principle, 
and one plate on intersections and developments. 

After you have penciled the next plate you should care- 
fully read the articles on inking, shading, dimensioning, 
and sectioning, which follow the description of the plate, 
before you attempt to ink in your drawing. 

LINES USED ON DRAWINGS. 

50. In the preceding articles we have already referred 
to four kinds of lines which are used on drawings, each hav- 
ing its own particular function to perform. These lines 
were used for the outlines of the objects, for the representa- 
tion of hidden parts, for indicating the centers of objects, 
and for projecting points or lines from one view to another. 
But a complete working drawing requires, besides those 
mentioned, lines for the following purposes: shading, which 
gives the workingman a clearer idea of the shape of the ob- 
ject, and dimensions, which give him the sizes of the various 
parts. 

Our students are to use the following lines, for the pur- 
poses indicated, on all drawings in this course, and a correct 
use of them will avoid any possible complications or errors. 



60 GEOMETRICAL DRAWING. 

The line ab, Fig, 65, is a light full line and is used for 

the outlines of figures. It 

should not be too light nor 

a b too heavy, but the widths 

c f/ shown in the figure should 

e f guide the student in draw- 
er h iug any of the lines. 

f j The line cd is a clotted 

line and is used for the 

FlG 65 representation of hidden 

parts on the object, that 
is. parts which the eye cannot see by looking at the object 
from the direction which gives the view we are drawing. 
The dots should all be of the same length and about ^ in. 
long. 

The line ef is a broken line, consisting of single dots 
and dashes. It is used for center lines of objects, as pre- 
viously explained. The dashes should be about J inch in 
length, with a dot between them. 

The line gh is a broken line, consisting of single 
dashes and two dots between them. It is used for pro- 
jection lines, as shown in Fig. 64. The dashes are about i 
inch long, and the distance between two dashes is about T V 
inch, in which there are two dots. 

The line ij is a broken line and consists of dashes. It 
is used for dimension lines, and the dashes should be from f$ 
to | inch long. 

The line kl is a heavy full line aud is used for shading 
the outlines of the figures, as will be explained later. 

The sizes of the dashes given above can be varied to suit 
the size of the various parts, making them smaller for very 
small parts, and even increasing their Length for very large 
drawings. 

PLATE VI.-TITLE: PROJECTION. 

."> 1 . Alter the student has satisfactorily completed the 
five plates of geometrical problems and has read the fore- 
going article on Projection, he can begin to draw Plate VI., 



GEOMETRICAL DRAWING. 61 

which illustrates the principles of projection of simple objects. 
A copy of this plate, on a reduced scale, will be sent to the 
student, and he is to follow the scales and dimensions marked 
on the sample plate. The size of the sheet is the same as for 
the other five plates, and the working edges are the same 
distance from the cutting edges as before. The space within 
the working edges, however, is not divided into rectangles, 
as was done in the previous plates, as the arrangement of 
figures on a drawing depends upon their size and number, 
and should be such as to be pleasing to the eye and so as to 
be easily understood by the workingman. If possible, no part 
of the drawing should be nearer than f inch to the working 
edges, and the arrangement should be such as to waste as 
little space on the paper as is consistent with clearness. 

Leave sufficient room above the drawing for the title and 
scale, and print your name and date below the lower work- 
ing edge, as shown on the sample plate. 

After reading the description of each figure, carefully 
studying the sample plate and understanding the significance 
of every line, you begin to pencil the drawing. Then, before 
inking the drawing, read the articles on inking, shading, 
dimensions, and sectioning, following this article, as the 
order in which a drawing is finished, is an important factor 
in the rapidity and exactness of its execution. 

52. Fig. 1 represents a rectangular (rect-an'gu-lar) 
prism 2|" long, 1-J-* wide, and f" thick. The front view or 
elevation, showing the prism standing on one of its small 
faces, with one of the largest surfaces towards the observer, 
is drawn first. Draw the horizontal lines ck and am of in- 
definite length 2f " apart, and join them by the lines ac and 
bd IV apart. These lines are projected upwards, and the 
lines eg and fh are joined by the horizontal lines ef and gh 
i* inch apart, completing the top view of the prism. To 
draw the side view, join the lines ik and hn by the lines il 
and km. f" apart. The line il represents the front face of 
the prism, and km the back. 

Fig. 2 represents a cylinder 1-J" in diameter and 2" long, 
standing on one of its circular bases. As has been stated 



62 GEOMETRICAL DRAWING. 

before, every circular or symmetrical object requires for its 
correct representation center lines which pass through the 
centers of the drawing and join the various views to each 
other. These center lines are the first to be drawn, and are 
marked AB and CD in the figure. From the point 0. where 
these two intersect, as a center, describe a circle having a 
diameter of l¥. This will be the top view or plan of the 
cylinder. To draw the front view of the cylinder, draw the 
two lines eg and ah of indefinite length. Join these by the 
lines ca and db, 1|" apart, by projecting the points i and k. 
The side view, being the same as the front view, is drawn in 
a similar manner, eg and fli being the projections of the lines 
cd and db. 

Fig. 3 represents a hexagonal (hex-ag'o-nal) prism stand- 
ing on one of its bases. The distance between the opposite 
corners of the hexagon is 1^", and the height of the prism is 
2". Draw the center lines AB and CD, and from the point 
0, where these intersect, as a center, describe a circle having 
a diameter of H". Within this circle draw a hexagon whose 
upper and lower sides are horizontal lines. To draw the 
front view, draw the two lines eg and ah of indefinite length, 
2" apart, and join them by the lines ca, pr, ms, and db. by 
projecting the points i, n, j, k. The side view is drawn by 
first drawing the center line T V and laying off on each side 
of this the distances Te and Tg, each equal to BO. Draw 
the lines ef and gh, which completes the side view. The 
student's attention is called to the difference between the 
front and side views of a hexagonal prism. 

Fig. 4 represents a cone, 2" in height, standing on its cir- 
cular base, which is IV in diameter. Draw the center lines 
AB and CD, and, with the point of intersection of these lines 
as a center, describe a circle IV in diameter. This will be 
the top view or plan of the cone. To draw the front view, 
draw the line ag of indefinite length, and on that line lay off 
the points A and C by projecting the points i and k. 2 above 
the line ac, and on the center line AB, locate the point b. the 
vertex of the cone, and draw the lines ab and cb. The side 
view is the same as the front view, and is drawn in a similar 



GEOMETRICAL DRAWING. 63 

manner, the point / being the projection of the point b on the 
center line FH. 

The student will observe that the front views of cylinders 
and cones are the same as the side views, and in practice the 
latter are always omitted. 

Fig. 5 represents a pentagonal (pen-tag'o-nal) pyra- 
mid standing on its base. The pentagon is inscribed in a 
circle 1-J* in diameter, and the height of the pyramid is 2 " . 
Draw the center lines AB and CD, and with the point O, 
where these intersect, as a center, describe a circle 1-J" in dia- 
meter. Within this circle construct a pentagon (by the method 
shown in Problem 16, Plate III.), the side de being horizon- 
tal, and the upper corner g being located on the center line 
AB. This completes the plan or top view. To draw the 
front view, draw the base line al of indefinite length, and on 
it project the points v, d, e, and r. T above the base, 
and on the center line AB, locate the point b and draw the 
lines ba, bs, bt, and be. To draw the side view, first draw 
the center line EF, and to the right of it, on the base line, 
lay off the distance nl equal to go, and, to the left, nh equal 
to Ox. Lay off nk equal to rp, and project the point b on 
EF, giving the point i. Draw ih, ik, and il, which com- 
pletes the side view. 

Fig. 6 represents a wedge, which is formed by cutting 
the prism shown in Fig. 1 from c to b. It is drawn similarly 
to Fig. 1, by first drawing the triangle abc, which is the 
front view. Its base is \\" long and its height is 2f " . The 
top and side views of the wedge in this position are the same 
as those of the prism in Fig. 1 . 

Fig. 7 shows a hollow cylinder standing on one of 
its bases. The outside diameter is IV, and the diameters 
of the three hollow cylindrical portions are l~", J", and 1-J-" re- 
spectively. Draw the center lines AB and CD, and with 
the point where these intersect as a center, describe a circle 
1\" in diameter, and one %" in diameter representing the 
smallest hole. Also describe dotted circles J" and 1^" in 
diameter to represent the other two interior portions of the 
cylinder which cannot be seen by looking down on it. To 



GEOMETRICAL DRAWING. 

draw the front view, draw the lines bt and aiv of indefinite 
length. '2 apart, and join them by the lines 6c/ and dc by 
projecting i and k. As the entire interior portion of the 
cylinder is hidden from view when looking at the front of it. 
it will be represented by dotted lines. Draw the horizontal 
lines b'd' and a'c of indefinite length, f from bd and ac re- 
spectively. U below b'd and above ac draw the horizon- 
tal lines / i and eh respectively. Draw the dotted vertical 
lines egln. fhmp. b'a . dr. f e . and i'h . by projecting the 
inner full and the two dotted circles in the plan. Join the 
dotted lines by semicircles witji a radius ol as shown. 

Instead of drawing a side view of this object, which would 
be the same as the front view, we draw a section of it. which 
i< obtained by imagining the object cut along the fine AB. 
This view is the same as the front view, with the exception 
that all the dotted lines in the latter are full lines in the sec- 
tion. Draw the center line EF and make the section the 
same as the front view, with the above exception. The sec- 
tioning will be explained in Art. 55. 

All the figures up to this point have been drawn full size, 
that is. the dimensions of the object have been represented 
in their true lengths on the drawing. The remaining figures, 
however, are drawn to a scale of 3 to 1 ft., or J size, and 
that part of the scale representing this relation should be 
used. The manner of using the scale has been explained in 
Art. 20. 

Pig. 8 represents a square, cast-iron washer of the 
dimensions shown on the drawing. First draw the center 
lines AB and CD. 5 above and below CD draw the hori- 
zontal lines bd and ac respectively, and join these by the 
vertical lines rib, 5 to the left, and cd, 5 to the right of AB. 
This will give a square each side of which is 1m . In a simi- 
lar manner draw a square within the larger one. each side 
of which is 4; . and one within this one. each side of which 
Using the point of intersection of AB and CD as a 
center, describe a circle \\ in diameter. This completes the 
top view or plan. To draw the front view, draw the base 
• of indefinite Length, and | above it the line gh. Join 
these by the lines ge and hf by projecting a and c. Draw 



GEOMETRICAL DRAWING. 65 

the horizontal dotted line ns 1^" above ef, and ik \" above 
ns. Draw the dotted lines In and os by projecting b' and d' , 
and the dotted lines ru and tv by projecting the circle in the 
plan. Locate the points i and A; by projecting a' and c', and 
draw vertical lines through * and k meeting the line ns 
(prolonged) . Join these points of intersection with g and h, 
and join these lines with those through i and k by small arcs 
as shown. 

Fig. 9 represents a 90° pipe elbow drawn to a scale of 
3" = 1 ft. As will always be done hereafter, the two views 
will be drawn together — that is, one view is not to be com- 
pleted before the other one is begun. First draw the horizontal 
lines abop and ders of indefinite length, J " apart, and taking 
any point, c, on the line abop as a center, describe the center 
line E A with a radius of 8". Draw a vertical line gf through 
the point c, of indefinite length, and Y to the left of it, the line 
ih. From A, where EA cuts gf, continue the center line 
EA by drawing the horizontal line AB, and at a convenient 
distance from A draw the vertical center line CD, intersect- 
ing AB at 0. From O as a center describe the following 
circles : One with a diameter of 1 U/ for the exterior of the 
flange, one with a diameter of 9£" for locating the centers 
of the holes in the flange, a dotted circle with a diameter 
of 6" + J" + f " = 7^" for the exterior of the pipe, and one 
with a diameter of 6" for the interior of the pipe. Where 
these circles intersect the center line CD, points will be found 
which are projected on the line gf, giving the points 
g, o' , I, n, if, f. With c as a center, and radii equal to 
co ', cl, en, and ct' , describe quadrants o'n', Ik, mn, and t'u' ; 
Ik aud mn being partly dotted, as shown. Join the two 
outer quadrants to the inner faces of the flanges by small 
arcs with \" radius, as shown in the figure. On the circle, 
with a diameter of 9J-" in the front view, step the radius off 
six times, beginning on the line CD. With these points as 
centers, describe six circles Y in diameter. Project these on 
the line gf and draw the horizontal lines joining gf and ih. 
Two of the holes are indicated by full lines and two by 
dotted lines, as part of the elbow is shown in section, the 
section being taken through CD. With c as a center, and 



GEOMETRICAL DRAWING. 



radii equal to the distances from c to the points just located 
on <ij\ mark similar points along the line ab. Join these 
points by vertical dotted lines to de. Draw the center lines 
tor the holes in a similar manner. Draw the lines tu 
i I . and vw. Join i ii and r//- to rs by fillets (arcs), as shown. 
Draw the vertical lines ro, *p, and the dotted lines represent- 
big the holes in the Mange rosp. The lines xy and ./•//' are 
drawn free-hand, and show how the front part of the pipe is 
broken away. 
Fig. 10 represents a connecting (con-nect big) rod 

drawn to a scale of 3* = 1 ft., a side view being shown below 

and a top view above. Draw the center lines AB and CD. 

Draw bd and ac equally distant from the center line. 3£" 
apart, and fh and eg 64- " apart. Equally distant from ('/>. 
%\ apart, draw the lines b'd' } a'c', fh. and eg'. Draw 
the vertical line aba'b', and »i; "from it ede'd'j is! to the 
right of this draw the line efe'f, and 0.1" from this the 
line ghg'h'. Draw the central part in the top view by 
drawing two horizontal lines 1 | apart, joined to the two 
ends just drawn by fillets. The side view of this is 
drawn by measuring off 2f* on cd, -1 of it on each side of 
AB, and \\ on fe. Join these points by lines, which are 
joined to the two ends drawn first by fillets, as shown. 
Draw the center line EFE F , and from the point where 
this intersects AB as a center, draw a circle | in dia- 
meter. \l to the left of /•;/«'. and 2f* to the right of it. 
draw center lines for two circles i" in diameter. The three 
circles just drawn are projected, as explained before, the g 
hole only penetrating the object as far as the key way. which 
is to be drawn next Draw the vertical dotted line ij \ t<> 
the left of EF; lay off the point / \ to the right of EF on 
ac and k i to the right of EF on bd. Draw the dotted line 
Ik. On the top view draw the horizontal lines i'k and ./'/ . 
equally distant from CD, I apart. .Join them by the fines 
i'j . o 8 , and k / . 1>\ projecting /. /. and L\ The construction 
at the right end being very similar to that at the left. QO 
further description is necessary. The key way is u at the 

top. and the center line -J,", from gh. 

Tin- completes the description of the figures on this plate. 



GEOMETRICAL DRAWING. 6? 

which should not be inked in until the following articles 
have been read. 

53. Rules for Inking-. As has been stated before, 
the order in which a drawing is made has a great deal to do 
with its neatness and the rapidity of its execution. The 
student should re-read Articles 33 and 50, and in inking the 
drawing follow the order given below, which has been found 
by experience to give the best results: 

1. Circles. 2. Arcs. 3. Irregular curves. 4. Horizontal 
lines. 5. Vertical lines. 6. Oblique lines. 7. Clean the 
drawing lightly. 8. Horizontal center lines. 9. Vertical 
center lines. 10. Shade lines. 11. Section lines. 12. Di- 
mension lines. 13. Dimensions. 14. Title and name. 15. 
Clean drawing. 

The student should never depart from this order, and he 
will find that the close observance of these rules will give 
him satisfactory results. Again, let us remind him to have 
the pen-leg of his compasses perpendicular to the paper; to 
stop the arcs at the proper place; to observe the rules about 
drawing irregular curves; not to have the ink in his pen 
give out before a line is completed; to draw all vertical lines, 
even the working edges, with a triangle; not to erase his ink 
lines when cleaning the drawing; to use the right lines for 
the purposes for which they are intended; to read the articles 
on shading, sectioning, and dimensions, which follow; to 
make his numbers and letters neatly and space the title 
properly; not to put the reference letters on the drawing- 
used in describing the figures; to clean his drawing after it is 
completed; and to write his name and address in pencil on 
the back of the sheet before he mails the drawing to us. 

54. Rules for Shading". Shade lines are used on 
drawings to aid the workingman in getting a clearer idea of 
the shape of an object by looking at the drawing, and to im- 
prove the general appearance by making the objects appear 
as they do with their natural lights and shades. Shading 
outline drawings is not a universal custom, and in some draw- 
ing rooms all drawings are shaded, while in others shading 
is omitted entirely. We therefore deem it necessary for our 



GEOMETRICAL DRAWING. 

students to have a knowledge of the principles of shadi] 
thai they may be in a position to apply them when required. 
As many authorities differ regarding the purely conventional 
method of shading drawings, we have adopted thai Btyle 

which is by far the simplest to understand, and the one 
which gives the most natural appearance to the drawings of 
the objects. 

We assume the lighl to come from the upper Left-hand 
corner of the drawing, parallel to the plane of the paper. 
making an angle of 45 with all horizontal and vertical Lines. 
This direction of the Light is assumed to remain the same for 
all views, and does not vary its direction whenever the ob- 
ject is drawn in a different position. By imagining a series 
of parallel lines making an angle of L5 with all horizontal 
and vertical lines to strike the object, we can determine those 
surfaces that are touched by these lines and those that are 
not. The former are called light surfaces, and the Latter 
dark surfaces. 

While no definite rules can be given for shading very com- 
plicated objects, or objects in oblique positions, we shall lay 
down the following rules for shading, which, however, can- 
not be strictly adhered to. and may even be violated for the 
sake of appearance in some of the drawings in this course: 

1. All edges formed by the intersection of two dark sur- 
faces are shaded, that is, the lines are to be drawn heavy. 

'!. All edges formed by the intersection of a light and a 
dark surface are shaded. 

■I All horizontal base lines are shaded. 

4. All horizontal top Lines are light. 

5. All right-hand vertical lines whose projections are 

perpendicular to horizontal lines in the plan or elevation are 

-haded. 

G. All left-hand vertical lines whose project ions are perpen- 
dicular to horizontal Lines in the plan or elevation are light. 

', . All sin-faces parallel to the paper are light surfa< 

Referring to Drawing Plate, Title: Projection, the above 

rules are applied in the following manner: 

Rule 1. The line />/ in Fig. 5 and c <■ in Kig. B. 
Rule 2. The line ms in Kig. 3 and he in Fig. 6. 



GEOMETRICAL DRAWING. 



69 



Rule 3. The line a b in Fig. 1 and op in Fig. 9. 

Rule 4. The line cd in Fig. 2 and fh in Fig. 10. 

Rule 5. The line dc in Fig. 7 and hf in Fig. 1. 

Rule 6. The line ab in Fig. 8 and ef in Fig. 10. 

Rule 7. The surface abed in Fig. 1 and rspm in Fig. 3. 

To further illustrate the use of these rules and their appli- 
cation to the shading of circles, let us refer to Fig. 66. 

ABCD is a square 
block with a square pro- 
jection or boss at a, a 
square hole at b, a cir- 
cular boss at d, a circu- 
cular hole at e, and a 
circular boss with a 
square hole at c. The 
arrows show how the 
light strikes the various 
surfaces, exterior and in- 
terior, and at a glance a 
person can see why the 



figure 
shown. 



is shaded as 




While this figure il- 
lustrates the shading of 
horizontal and vertical 
lines, Fig. 67 shows 
how oblique lines are 
shaded. In this figure . 
we have six triangular 
pieces, raised above the 
' paper, the interior of 
them being cut out as 
shown. The central 
part is a hollow cylin- 
der, also raised above 
the paper and above 
the triangular pieces 
just mentioned. 
The arrows indicate the direction of the light, and the 




70 



GEOMETRICAL DRAWING. 



student will notice that it is tangent to the four circles at the 
points where the line AB cuts the circles. This line makes 
an angle of 45° with the horizontal center line CD. He will 
also notice that the piece bounded by the arc kl has its two ex- 
terior edges shaded and its two interior ones light: the reverse 
is the case with the piece bounded by ef. This is a case which 
does not arise frequently, but shows the necessity of consid- 
ering each surface separately before shading the edges which 
bound them. 




Fig. 68 illustrates the method of shading circles. The 
lines AB and CD make angles of 45° with the horizontal. 
The light comes in a direction parallel to CD and illumi- 
nates the upper half of the ring, and does not reach the 
lower half, that is, the part below AB. The reverse takes 
place with the inner circle, and the shaded portion of this 
circle will therefore be above the line AB. In other words. 
the rays of light e'e, h'h, and/'/, g'g are tangent to the cir- 
cles at the points where AB cuts the circles. 

To shade the circles, the center o is moved along the line 
CD in the direction s or t, and, with the same radius that 
was used for describing the original circle, part of another 
circle is described on that side of AB which is to be shaded. 
This second circle will meet the original one at the points e. 
/'. (/. and //. and care should be taken not to extend it beyond 
the point of coincidence. The width of the shaded line is 
determined by the amount the center is shifted. 



GEOMETRICAL DRAWING. 71 

In shading concentric circles, such as are shown in Fig. 
68, in which the distances ab, ef, cd, and gh are small com- 
pared with the diameters of the circles, it is very essential to 
move the center in such a direction as to keep the above dis- 
tances equal. That is, for shading aedh the center should 
be moved in the direction t, so that the heavy line is on the 
outside of the circle; and for shading bfcg the center is 
moved in the same direction, so that the heavy line is on the 
inside of the circle. Fig. 69 shows how distorted the ring 
will appear if the center is moved in the direction s. In 
shading very small circles the heavy line should be put on 
the outside, so as to keep the interior space circular. How- 
ever, the rule regarding the shading of concentric circles 
takes precedence of this one. 

55. Sectioning*. In the description of Fig. 7 on the 
plate entitled "Projection," we referred to the use of sec- 
tions and how they are obtained. All parts of the object 
which are cut by the imaginary cutting plane are sectioned 
or filled out with section lines. These are drawn with the 
triangle at an angle of 45°. They are not penciled, but are 
inked in after the outlines of the figures have been drawn in 
ink and the drawing has been shaded. They are spaced by 
the eye and are a full T y apart, or about three spaces to i". 
For larger drawings the lines are drawn T y apart. By 
means of sectioning we can further indicate the number of 
parts on an assembly drawing, as the different parts are sec- 
tioned in opposite directions. The material of which each 
part is made can be shown by adopting different styles of 
section lines. For cast iron, single light lines at an angle of 
45°, the above-mentioned distance apart, are used. The 
styles adopted for sectioning other materials will be taken up 
in the next paper on drawing. 

56. Dimensions. We may say that the most impor- 
tant element on a working drawing is the correct placing and 
completeness of its dimensions. If the workingman only 
had a rough free-hand pencil sketch to work from, he could 
construct the object, if all the dimensions were properly 



72 GEOMETRICAL DRAWING. 

marked on the sketch. The student should therefore pa}' 
particular attention to this portion of his work; and while he 
will only be expected to copy the dimensions on these plates, 
he should study the method employed, so that he will know 
how to place dimensions on working drawings later in his 
career. 

No definite rules about the number and position of dimen- 
sions can be given, but in general we may say, insert all the 
dimensions that are required by a workingman or builder in 
order to construct the object by having reference to the 
drawing only, to place them where they are most liable to 
be looked for. to make the figures plain, to put the ' i over-all" 
dimensions on a drawing, and not to duplicate any dimen- 
sions. 

The dimension lines are a series of dashes, as explained in 
Art. 50 and illustrated in Fig. 65 by the line ij. The 
" over-all" dimensions, that is, those indicating the entire 
length of an object or a section of it, are generally placed on 
the exterior of the drawing, about -f^ to ¥ away from it. 
As will be seen by referring to the plate just drawn, the pro- 
jection lines are frequently used as the limits for the dimen- 
sion lines, as ec and fd in Fig. 1, or ds and cv in Fig. 7. The 
diameters of cylinders and cones can either be marked on the 
plan, as in Figs. 2, 4, and 7, or on the elevations. In the 
case of prisms and pyramids the diameters of the circum- 
scribed circles are given, as in Figs. 3 and 5. The diameters 
of holes may either be marked on the elevation, as the ¥ 
hole in Fig. 7, or within the circle in the plan, as the IJ" d 
hole in Fig. 8, or marked on the outside, as the ¥ holes in 
Fig. 9, or as the f " hole is marked in Fig. 1 0. It is custom- 
ary to draw center lines and refer the dimensions of other 
parts to these, as the location of the line EF in Fig. 10, and 
then marking the distance between it and the line ij \ . 
When the space is too small for putting in the dimension 
lines and figures, put the lines and arrow-heads on the out- 
side and the figures within the space, as the i " dimension of 
the line ig in Fig. 9, or the ¥ dimension in Fig. 10, just 
referred to. The radii of fillets are marked as in Fig. 7, 
front elevation, by writing 1 \-" r. or without giving the length 



GEOMETRICAL DRAWING. 73 

of the radius and simply locating the center, as is done in 
Fig. 10, near the points d, e, d', e'. 

Care should be taken to have the dimension lines touch 
the extension or limit lines, as is shown by the upper line in 
Fig. 70, and not like the lower line in the figure. 

The arrow-heads should [not 
open as they do on the line cd, a p ^s 

but should be made neatly, as I , y* __ 1 

" they are on ab. They, as well ' Fm 70 ' 

as the figures, are made with a 
Gillott No. 303 pen. 

The figures should be legible and -fj" high. The frac- 
tions are -f^" high, as shown in Fig. 22, and the horizontal 
dividing line, which should always be used in a fraction, is 
in line with the dimension lines. The numerator and denomi- 
nator are placed under each other in a slanting direction, 
the slant being the same as that of the whole numbers. 

PLATE VII.— TITLE: PROJECTIONS AND 
CONIC SECTIONS. 

57. The objects drawn on Plate VI. were so placed that 
the center lines, and at least one flat surface of each, were 
parallel to the plane of the paper. The placing of objects in 
this convenient position for drawing them is only possible 
when the objects are small, or if the details of a large object 
are drawn separately: but when it is required to make a 
drawing of the assembled object, the various parts may be 
located in such positions as to have their center lines make 
angles with the plane of the paper and have none of their 
surfaces vertical or horizontal. In order to prepare the 
student for these cases, the objects drawn on this plate are 
all placed, or cut by planes, at an angle, and this gives rise 
to certain constructions not necessary on Plate VI. The 
principles involved in this plate, and the one preceding and 
following it, should be thoroughly understood by the stu- 
dent, as they are fundamental and will be continually 
applied in the plates which follow. 

Fig. 1 represents a rectangular prism 21" long, 1\" 



74 GEOMETRICAL DRAWING. 

wide, and J* thick, the 1-j" sides making an angle of 20° with 
a horizontal line. There is a projection on one side of the 
prism, as shown. The front elevation is drawn first, because 
in that view only, are all the lines drawn in their true 
lengths. The drawing is made full size. Draw the line 
ad of indefinite length, making an angle of 20° with a 
horizontal line ap'. From the point a, where ad inter- 
sects the horizontal line, draw the line ab perpendicular to 
ad of indefinite length. Draw cd parallel to this line and 
1\" from it. Join these lines by the line be, parallel to ad 
and 21" from it: \ n from the line be and parallel to it, draw 
the line ef. Draw^ gh parallel to ef and f* from it. Join 
these lines by fh parallel to cd and \" from it. This com- 
pletes the front elevation. To draw the plan, draw two 
horizontal lines, im and jn, f" apart, and join them by the 
lines ij, kl, win, obtained by projecting the points b, c, and d 
respectively, on the line ikm; \" from im draw the horizon- 
tal line ot and the line pu parallel to, and f from ot. 
Join them by the lines op, rs, and tu, obtained by pro- 
jecting the points e, f, and h on the line ot. To draw the 
side view, first draw the vertical lines a'c', g'f, u't', and 
p'o' the given distances apart. Then join these by horizon- 
tal lines as shown, by projecting the points a, d, g, It. f. 
and c. 

Fig. 2 represents a full-size drawing of a hexagonal 
prism having two of its parallel sides parallel to the plane 
of the paper, the base of the prism making an angle of 20° 
with the horizontal. Draw the line ad, making an angle 
of 20° with the horizontal; 2" from ad and parallel to it, 
draw the line be of indefinite length. From the point where 
ad intersects the horizontal, draw a line dc perpendicular to 
it. Draw the center line AB parallel to dc and |* from it. 
From the point /, where AB intersects the line be. as a cen- 
ter, and with a radius of | ". describe a semicircle cutting the 
line be in b and c. Divide the semicircle into three parts, 
hi. ij. and/, by stepping off the radius three times around 
the semi-circumference. Draw the lines ./>/. ie, and ba 
parallel to AB. To draw the top view or plan, draw the 
horizontal center line CD and the vertical center line EF hy 



GEOMETRICAL DRAWING. 75 

projecting the center I. From the intersection of CD and 
EF, lay off the distances or and os equal to if or jh, and 
through these points draw the horizontal lines tu and vw of 
indefinite length. The points n, v, t, in, y, p, iv, u, and k 
are obtained by projecting the points a, e, b, f, h, and c 
respectively. Join the points thus obtained by lines as shown 
on the drawing. This completes the top view. To draw the 
side view, first draw the center line GH. Lay off a distance 
on each side of it equal to or or os and draw the vertical 
lines f'g' and i'y'. On these lines and on GH locate the 
points b',f, i\ h', e', d , g ', y' , and d'hj projecting the points 
b, f, h, c, g, and d respectively. Join the points by lines as 
shown in the figure. 

Fig. 3 represents a full-size drawing of a cylinder, 1-g-" 
long and 1\" in diameter, whose base makes an angle of 20° 
with the horizontal. It is drawn similarly to Fig. 2 by first 
drawing the lines ad, be, cd, and AB. Then describe the 
semicircle befc and divide it into any convenient number of 
equal parts, with a 60° triangle, six in this case. Project 
these points on be parallel to AB and obtain the points b, i, 
j, k, I. Draw ba, which completes, the front view. To draw 
the top view or plan, draw the center lines CD and EF and 
lay off om and on equal to h A, or %" , and draw the horizon- 
tal lines r'm and p'n. Also draw the horizontal lines j'k' at a 
distance from CD equal to je or kf; also i'V , equal to is or 
Ig. Project the points b, i, j, k, I, c, upward, and these in- 
tersect the lines just drawn at the points b', i' , j', k', V, and 
c' . Through these points draw a curve with the irregular 
curve. Project the point t on the lines r'm and p'n, and a on 
the line CD, and through r', a', p' draw a curve in the same 
manner as mb'n was drawn. This completes the top view. 
To draw the side view, first draw the center line GH, and 
draw the vertical lines rt' and pt' of indefinite length, at a dis- 
tance equal to om or on on each side of it. Draw the center 
line KL by projecting the point h. Draw the lines e'e' and 
g'g' at a distance from GH equal to that of the point k' from 
CD, and d'd' and /'/' equal to the distance of the points V 
from CD. Project the points b, i, j, k, I, c to the right, and 
where these lines intersect the lines just drawn, we get the 



76 GEOMETRICAL DRAWING. 

points e, d ' , ri , f, g' , and m'. Through these, draw an 
ellipse with the irregular curve, and in the same manner 
draw a curve through the points t'd't' obtained by projecting 
the points t and d. 

Fig. ± represents a full-size drawing of a truncated 
(trim ca-ted) pentagonal pyramid standing on its base, 
the cutting plane making an angle of 30° with the horizon- 
tal. Draw the center lines AB and CD, and from the point 
o, where these intersect, as a center, describe a circle with a 
radius of \%". "Within this describe a regular pentagon, the 
lower side of which is a horizontal line. Draw the horizon- 
tal line al of indefinite length and on it project the points a', 
h ', g'j and c , giving the points a, h, g, and c. Join these 
points with the vertex b, located on the center line AB, %\" 
above al. by drawing the lines ab, lib. gb, and cb. Draw 
the line dj, representing the cutting plane, cutting AB at a 
point i" above the base line, and making an angle of 30° 
with the horizontal. The line dj cuts the lines just drawn in 
d, e, f, i r and j. Project these points on the lines oct', ob ', 
oc', og', and oh', and join the points d', f',j', i' , and e'. This 
completes the top and front views. To draw the side view, 
first draw the center line EF perpendicular to al. Project b 
on EF ; make vk equal to ok', vl equal to ob', and vm equal 
to m'c , and draw kt. mt. and It. On these lines project the 
points d. e, f, i, and j, and join the points p, r. s, n, o by 
lines as shown in the figure. 

Fig. 5 represents a full-size drawing of a 90° square 
elboAV standing on one of its legs. The top and front 
views at the left show the elbow in a position with the front 
surface parallel to the paper, this surface making an angle 
of 25° with the horizontal in the views at the right. The 
front view at the left is required in order to draw the front 
view at the right, and the top view at the left is reproduced 
at the right, being placed at the given angle. Draw the line 
an of indefinite length, fe f l ' above it, on&cd, 1 above that. 
Then draw a 'e' and c'd , 1" apart, and draw the vertical lines 
aca'c. bfb'f, and ede'd', f and |* apart respectively . Draw 
the dotted lines /'/. ih\ (j>/.(/lt. ij . fk', and i '//', yV," from the 
lines parallel to them: draw g h by projecting the line gh. 



GEOMETRICAL DRAWING. 77 

To draw the views at the right, first redraw the top view 
at the left by placing it in the position Vw't'v ', the line i'v 
making an angle of 25° with the horizontal. The front view 
at the right is obtained by projecting the horizontal lines of 
the front view at the left for the horizontal lines and the 
points on the right-hand top view for the vertical lines, as 
the projection lines on the drawing clearly show. 

Figs. 6, 7, and 8 represent what are called conic (con'ic) 
sections. These are figures obtained by cutting a cone by 
a plane. If a cone is cut by a plane through vertex and base, 
the section will be a triangle, and if cut by a plane par- 
allel to its base, the section will be a circle. If an oblique 
plane cuts the cone above its base, the section will be an 
ellipse, as shown in Fig. 6. A parabola (par-ab'o-la) is a 
figure obtained by a plane cutting a cone parallel to one of 
its sides, as in Fig. 7. An hyperbola (hy -per bo-la) is 
obtained when the cutting plane makes any angle with the 
base greater than that made by a side of the cone (see Fig. 8). 

Fig. 6 represents a full-size drawing of a cone which has 
been cut by a plane making an angle of 30° with the horizon- 
tal. Draw the center lines AB and CD, and from o, their 
point of intersection, as a center, describe a circle with a 
radius of f ". Draw the line ag of indefinite length, and on 
it project the points a' and c '. Join a and c with a point b, 
on AB, 2i" above ac. Through i on the line AB, draw the 
line de, $" above ac, making an angle of 30° with the hori- 
zontal. Divide the circle in the top view into twelve equal 
parts with the 60° triangle as shown, and draw radial lines 
to the center, o. Project the points on the circumference on 
the line ac, giving the points j, k, etc. Draw lines from 
these points to the vertex b, cutting the line de in the points 
n, o, r, and s. Project these points up, cutting the radial 
lines in the points d', ri , o' , r' , s' , and e\ and make oi' equal 
to il. Through these points draw an ellipse with the 
irregular curve. 

To draw the side view, first draw the center line EF, and 
on it project the point b, giving the point h. Make fg equal 
to ac, and draw fh and gin. Lay off vu and vm equal to the 
perpendicular distances from the center line CD to u' and m' 



78 GEOMETRICAL DRAWING. 

respectively, in the plan, and lay off the same distances on 
v g. Draw lines joining these points with the point h. Pro- 
ject the points i, o and r, n and s, d and e, on the lines fh, 
mh, uh, and vh respectively, continuing these projection 
lines till they cut similar lines on the other side of vh. 
Through the points i" , o" , r" , n" ' , s", d" , e", etc., draw an 
ellipse with the irregular curve. 

Fig. 7 represents a full-size drawing of a cone cut by a 
plane parallel to one of the sides. First draw the center 
lines AB and CD, and with their point of intersection o, as a 
center, describe a circle 1J" in diameter. Draw the line ak of 
indefinite length, and on it project the points a' and c. Join 
a and c with the point b on the line AB, %\" above ac. Draw 
the line ei parallel to the side be, cutting the center line AB 
at h, W" above ac. Divide the circle in the plan into twelve 
equal parts with the 60° triangle, as shown, and draw radial 
lines to the center o. Project the points on the circumfer- 
ence on the line ac, giving the points /, g, etc. Draw lines 
from these points to the vertex b, cutting the line ei in the 
points r and s. Project the points e, r, and s up, cutting the 
radial lines in e , r'. and s', the projection of the point i cut- 
ting the circle in the points i'. Make oh' equal to hj in the 
front elevation and join the points i' , h' , s', r, and e with 
an irregular curve, i'i' being a vertical line. To draw the 
side view, first draw the center line EF, and on it project 
the point b, giving the point /. Make dk equal to ac and 
draw dl and gl. Make vw, vn, and vu equal to the per- 
pendicular distances from the center line CD to the points w . 
n', and i respectively, repeating this construction on the line 
vg. Join these points with the vertex I. Project the points 
e, r, s, and h on the lines Iv, Iw, In, and Id respectively, 
giving the points e", r" , s", and h". Through these points 
draw a parabola with the irregular curve, terminating in 
the points u and x on the line ag. 

Fig. 8 is a full-size drawing of a cone cut by a plane making 
a greater angle with i\w base than that made by the sides. 
The plane is parallel to the axis AB. Draw the plan, trout 
and side elevations of the cone as in Figs. r> and I. the dia- 
meter of the base being H", and the height 2±". Again 



GEOMETRICAL DRAWING. 79 

divide the circle into twelve parts; project the points on the 
circumference on the line ac, and join the points, e.f, etc., 
with the vertex, b. Draw the vertical line ed, i" from AB, 
and project it upward, getting the line ef. To complete 
the side elevation, lay off the points *, n, and r as in Figs. 
6 and 7, and draw the lines hr, hn, and hi. On these lines 
project the points d, x, and u, and through the points d ' , 
x', u', and r, draw an hyperbola with the irregular 
curve. 

Fig. 9 represents a full-size drawing of a triangular 
prism with two projections of the given dimensions, 
the base of the prism being parallel to the paper, and the long 
edges making an angle of 20° with the horizontal. First 
draw the top view at the left by drawing the lines ac and bd 
If" apart, making an angle of 20° with the horizontal. Draw 
a b and cd, 2i" apart, and perpendicular to these lines. Draw 
ef midway between ac and bd and parallel to them. £" from 
ab, and parallel to it, draw the line gj; f" from this, the line 
ik, and f" from this, the line hi, all parallel to ab, and of in- 
definite length. Draw gh, -J" from bd and parallel to it; -A-" 
from this line on hi locate the point l; i" from ef and parallel 
to it draw the line mn, and i" from it and parallel to it the 
line op. Draw the line ftp parallel to, and f" from dc, andf " 
from it and parallel to it, the line om. This completes the 
top view with the exception of the lines jk and kl, which 
cannot be drawn until the point k is located. This requires 
the drawing of an end view. Draw the vertical line c'b' of 
indefinite length; \f" from it and parallel to it draw the line 
f'b" of indefinite length, and J-" from it, the line p' m' of inde- 
finite length. On the line f'b" project the points / and e 
and on b'c project the points c, d, and b. Draw b'b" , d'f 
and c'f. To project the point h, I, and n, we draw lines 
parallel to bd, which intersect fd in h' , f and n' and the line 
f'd' in h", I , and n". From these points draw lines parallel 
to b'd', giving the lines h"g ! , Vg" and ri'i. On these lines 
project the points g, i, h, and /, giving the points g', g". i\ 
s, s' f r, and /". Draw the lines cj'g" , g"i', is, ss', s'r, and 
rl". Project the point I on ik, giving the point k, and draw 
the lines jk and Ik. Project the points m. n, and p on the 



80 GEOMETRICAL DRAWING. 

linep'm', giving the lines m m\ k'k", and pp ". Draw the 
line k"v parallel to d'f. 

Fig. 10 represents a \vr ought-iron jaw of the given di- 
mensions drawn to a scale of 6" = 1 ft. To draw the front 
view, first draw the center line EF, making an angle of 20° with 
the horizontal, and AB perpendicular to it. Draw ab parallel 
to, and 1" below EF, Is and rp, f -f- 2 = f" on each side of EF 
and parallel to it; cd parallel to, and 2J" above ab, ef parallel 
to, and 1" above cd, and gh parallel to, and 1" above ef. 
These lines are joined by ge and hf, ea and fb, ci and dj, 
parallel to AB, and If", 2i", and 1±" apart respectively. 
Round the corners by small arcs, as shown. To draw the 
side view, first draw the vertical center line CD, and parallel 
to it draw the lines v'f, 2" apart. Project the points/ and o, 
and draw the lines o'f lj-" apart; draw the arcs at the points 
/'. To draw the curves olio', v'b'v', v'j'v', v'C'v', i'e'g' , 
etc., we make use of constructions similar to the one shown 
in Fig. 3. Draw the quadrants hk and r's', divide each 
into any number of equal parts — three in this case; project 
the points on the arc hk on oh, giving n and m. Project o, 
n, m, and h to the right, and the points on the arc r's' up- 
ward. Where these intersect, we get the points h',m', n', and 
o', and through these points the curve h ' ni'ii'o' is drawn. A 
similar construction is used for the curves below, and the 
student should study the principle thoroughly, for it will be 
used very frequently in this course and in practice. 

Fig. 11 represents a base of a cast-iron coluuiii drawn 
to a scale of 1?" = 1 ft. To draw the front view at the left, 
we first draw the line ab, making an angle of 60° with the 
horizontal. Draw the center line AB perpendicular to ab, at 
a convenient distance from the point a. From the point o, 
where ab and AB intersect, as a center, and with a radius of 
9|-", describe a semicircle befa. Divide this intoo equal parts 
with your 60° triangle, and draw the semi-hexagon befa. 
Draw the line cd parallel to ab and '!" from it, Project the 
points e and /on aft, and draw the lines gi and mil. With 
o as a center, and radii of 11 " and 2J' respectively, describe 
two circles, about which two Bemi-hexagons are circum- 
scribed. Project the points//. /\ x,j f /. p, etc., perpendicularly 



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GEOMETRICAL DRAWING. 81 

on ab, and continue the lines to the points r, k, t, y, z, u, w, s, 
joining these points by the broken line rs. To draw the ribs, 
we complete the dotted plan to the left of ab by drawing a 
line parallel to ab, \" from it, and drawing the ribs between 
v and e, and x and /, 1" in width. Let us further illustrate 
the construction by referring to the rib r'b'n's'. Project the 
points r' , s', and n' on the line cd, giving the points u' and 
v' and the line t'f. Locate n' on the line n't, 6-J-" above cd, 
and obtain the point e' by projecting b'. To draw the end 
view at the right, first draw the center line CD. Parallel to 
it, and at a distance equal to om' on each side of it, draw the 
lines s"m" , the points s". and m" being found by projecting 
the points i and m. Project the points n, a, and c, locating 
the points n" , a' and c" , and draw the lines a'm", c"n" , 
and s"l" . Lay off 2J-" on each side of CD, equal to the dis- 
tance oo' in the front view, and draw the vertical lines l"h". 
Make part of CD a full line, as shown; also draw y'z' paral- 
lel to, and 1" from l"u". On these lines project the points r, 
k, t, y, z, u, w, s, and draw t"r", t"u" , u" s' , k'y' , y'z' , and 
z'w'. Project h' on h"l" and get the point h", and d' on n"c" 
and get d" . Project b' to the right and draw a line from d" 
parallel to c"n" (on the right side of CD), intersecting the 
projected line in b" . From b" draw a line parallel to c"n" 
(on the left side of CD). Project the point p' , and this is in- 
tersected by a vertical line from b" \np" . Join d" andp" by 
the line d" p" , and draw a line parallel to this from i' , the 
projection of i' on s"m" . Draw h" p" and i"d" . This com- 
pletes the projection of one of the ribs. The others are pro- 
jected in a similar manner, and the width of the one in 
front is 1" , as is shown in the front view. The broken part 
of the column is sectioned to represent cast iron. 

PLATE VIII.— TITLE: INTERSECTIONS AND 
DEVELOPMENTS. 

58. Fig. 1 represents a full-size drawing of three in- 
tersecting cylinders, the axes intersecting at right 
angles and at an angle of 45° respectively. First draw the 
center lines AB and CD, and with their point of intersec- 



82 GEOMETRICAL DRAWING. 

tion o as a center, describe a circle If" in diameter. Pro- 
ject this downward and draw the front view of the cylin- 
der abed, 2-j" in length. l£" above the base ac draw the 
center line GH perpendicular to AB, and draw the hori- 
zontal lines eg and fh, £" apart, joined by the vertical 
line ef, f" from ab. From a point j\" above ac, on AB, 
making an angle of 45° with it, draw the center line EF. 
Parallel to this draw the lines ik andJZ, %" apart, jl being IV" 
in length. Join these lines by kl perpendicular to them. 
Again referring to the plan or top view, draw the horizontal 
lines e'm" and f'ni', Y apart, and draw e'f by projecting ef. 
Draw h'o' and n'o', £ " apart. The ellipse o"k'l'o" is obtained 
by the method shown in Fig. 3, Plate VII. , and is again 
shown in this figure. To find the curves of intersection 
between the cylinders abed and efgh, describe the semicircle 
e'yf and divide it into any number of equal parts — six in this 
case. Project the points on the semi-circumference to the 
left and get the points t" , u" , g ', n", and x". These pro- 
jection lines cut the line e'f in points which are reproduced 
on ef, giving t, u, v, w, x. Project these horizontally to 
the right and the points t", u" , g', n" , x" downward, and 
through their points of intersection g, t' , u' , m, w' , x', and 
h draw a curve. In a similar manner the projections of the 
points p, g, F, r, s cut the line kl in points which are 
reproduced on the perpendicular distance between the two 
lines n'o" and h'o" and are projected to the left, giving the 
points p", g", i' , v" , s". These are projected downward, and 
the points p, g, F, r, s are projected parallel to hi. and 
through their points of intersection, p', g\ n, r', s' 3 draw a 
curve, which will be the front view of the curve of intersec- 
tion of the largest and smallest cylindrical surfaces in Fig. 1. 
Fig. 2 represents the development of the cylinder efgh in 
Fig. 1. Draw the horizontal line ab equal in length to the 
circumference of the cylinder efgh, which is equal to the 
diameter I" X 3.1416 = 2.749', or nearly %$". Draw the 
indefinite vertical lines ac and bd and divide the line ah into 
as many equal parts as the circumference of the cylinder was 
divided in Fig. l -in this case twelve. Draw vertical lines 

through these points and make ac. jj ', and bd equal to rut in 



GEOMETRICAL DRAWING. 83 

the front view; ee' , ii', kk' , and oo equal to uu' or ivtv'; ff, 
hh', IV, and nn' equal to W or xx' ; gg' and mm' equal to eg 
or fh. Connect the points c, e', f, g', h\ etc., by using the 
irregular curve. 

Fig. 3 represents the development of the cylinder klij in 
Fig. 1. Draw the horizontal line ef equal in length to the 
circumference of the cylinder klij, or f" X 3.1416 = 1.96", 
which is very nearly equal to lfi". Divide this line into 
twelve equal parts and draw twelve vertical lines as in Fig. 
2. Make eg and fh equal in length to ik in the front view, 
and mn equal to jl. The distances on each side of mn are 
respectively equal in length to the perpendicular distances 
from the line kl to the points s' , r ', n, g' , p '. Through the 
points thus obtained draw a curve. 

Fig. 4 represents three cylinders intersecting- at 
angles of 45° and. 30°. Draw the vertical center line 
AB, db and ac parallel to it, 1-J" apart and of indefinite 
length. Lay off If" on AB, giving the points C and E. 
Draw the center lines CD and EF, making angles of 45° and 
30° respectively with AB. Draw be and af parallel to CD 
and make af ■£/ long. Draw ef perpendicular to CD. 
Draw dg and ch parallel to EE, making ch \\" long, and 
draw gh perpendicular to EF. Describe semicircles with o, 
o', and o" as centers, as shown, and divide each one into six 
equal parts, projecting these points on the lines gh, dc, ba, 
and ef. This completes the front view of the cylinders. 
As will be noticed, the curve of intersection of cylinders of 
the same diameter is a straight line. 

Fig. 5 shows the development of the three cylinders drawn 
in Fig. 4. The development of the end ba of the cylinder 
abed being the same as the end ba of the cylinder baef, but 
one curve need be drawn, the same being the case for the 
intersection dc. Draw ab equal in length to IV X 3.1416 = 
3.534', or 3Jf" nearly, and divide it into twelve equal parts. 
Draw the vertical lines ac, ee' , ff, gg', etc. Make ap, eg, 
fr, gs, ht, iu, jv equal to the distances from the line gh to 
the points d, p, h, E, i, j, c respectively, and reproduce the 
curve drawn through these points on the other side of the 
point v. Make pp', qq', rr', ss', W , uu' , vv' equal to the 



84 GEOMETRICAL DRAWING. 

distances ea, jn, ini, EC, hi, pk, and db respectively, and 
reproduce this curve on the other side of v'. Then make p'c 
equal to be in the front view and draw the horizontal line cd. 
The sections apbo, pp'oo', and p'co'd are the developments 
of the cylindrical sections dghc, abed, and beaf in the front 
view respectively. 

Fig. 6 represents a cone cut by a plane making an angle of 
30° with the axis of the cone. Draw the front view of a 
cone 2|" high, with a base whose radius is -J". Draw the 
line CD, making an angle of 30° with the center line AB, 
and 1" above the base, measured on the center line. \" 
above the point g, on the line CD, draw the line EF, which 
represents a plane cutting the cone parallel to the base. 
With o as a center, and a radius of £-*, describe the semicircle 
aAb. Divide this into six equal parts; project the points on 
ab, and through them draw lines to the vertex of the cone, 
cutting CD in c, e, f, g, h, i, and n. Project these points 
on the line bd, giving the points c', e', /'. g ', h', i', and n. 

Fig. 7 represents the development of the frustrum of the 
cone acbn drawn in Fig. 6, and also of the frustrum amrb, 
whose upper and lower bases are represented by mr and ab 
respectively. Draw the center line AB, and with the point 
o as a center, and a radius equal to the slant height bd of the 
cone, describe an arc. The length of this arc is equal to the 
circumference of the base of the cone, and is obtained by 
stepping off on it the chords of the arcs at, tu, uA, etc., 
twelve times. If greater accuracy is required, the semicircle 
and the arc would have to be divided into a greater number 
of parts, or the method shown in Problems 23 and 24, Plate 
IV., should be used. From the points of subdivision of the 
arc draw lines to the center o. Make aa'.cc', dd', ee', ff, gg', 
ii' equal bn, bi', bh', bg', bf, be', be' in the front view 
respectively, and reproduce the curve drawn through the 
points a', c', d' , e', f ', g' , and %' on the other side of the point 
i'. aa'bb' represents the development of the frustrum aben. 
From o as a center describe the arc nst with a radius equal 
to dr in the front view, anbt being a development of the 
frustrum al>)iu\ 

Fig. 8 represents i\ heptagonal pyramid cut by a plane 



GEOMETRICAL DRAWING. 85 

making an angle of 30° with the axis. Draw the center line 
AB, and about this line as an axis construct the pyramid 
abc of the given dimensions. Draw the line de, making an 
angle of 30° with the horizontal and cutting the center line 
at the point s, J" above the base, and the horizontal line mn 
cutting AB, f" above s. Project the points d, o, and r, 
which are the intersections of the lines ac, fc, and gc with 
de, on be, giving the points d' , o' , and r' . 

Fig. 9 represents the development of the frustrum abde of 
the pyramid abc drawn in Fig. 8, and also of the frustrum 
abmn. Draw the center line AB, and with o as a center, and 
a radius equal to ac, the length of one side of the pyramid in 
Fig. 8, describe an arc. On this arc step off the distance ap, 
the length of one side of the heptagon, seven times, and draw 
the lines ao, bo, co, do, eo, etc. On these lines step off aa' , 
bb', cc ', dd', etc., equal respectively to bd' , bo', br', and be in 
Fig. 8, and draw the lines a'b', b'c', c'd', d'e', etc., repeating 
the same construction on the other side of AB. aa'm'm is 
the development required. 

By laving off the distances af, bg, ch, etc. , on the radial 
lines, all equal to the distance bn in Fig. 8, and drawing the 
lines fg, gh, hi, etc., we obtain afhn, which is the develop- 
ment of the frustrum amnb. 

Fig. 10 represents a frustrum of a cone intersected by a 
cylinder, in plan and side elevation. First draw the center 
lines AB and CD intersecting at o. With this point as a 
center, describe' two circles 2i" and f" in diameter respec- 
tively. Draw the vertical lines ab and cd, 1-J-" apart, of 
indefinite length, and draw ac, 2-^" below CD. Draw the 
vertical lines ef and gh, 2f" apart, and on these project the 
extremities of the vertical diameters of the circles in the plan, 
obtaining the points e, f, g, and h. Draw the lines eg and 
fh. Draw the vertical center line EF, 1^" from gh, and 
a'm' and c'i', 1^-" -apart, of indefinite length, joining them by 
a'c' projected from the plan. Draw the semicircles aAc and 
a'Fc', having ac and a'c' as diameters respectively. Divide 
each semi-circumference into six equal parts and project these 
points upward by vertical lines of indefinite length. In the 
side view these lines intersect the centre line CD in the 




LIBRARY OF CONGRESS 

liflllillllllllUflWIl^ 
019 971 100 3 



86 GEOMETRICAL DRAWING. 

points m", v" , r", d", s", n", and i", and the line eg in m . p. 
t, u, to, x, and *'. With o as a center, and radii equal to 
m"m', v"p, r't. d"u, s" tv. n"x, and i"i', describe arcs cutting 
similar vertical lines in the plan in m, k and v, p and r, b 
and d, j and s, b and n, and i respectively. Join these 
points with the irregular curve, the part bmd being full and 
bid dotted. Project the points m, v, r, d, s, n, and i to the 
right, cutting similar vertical lines in the side elevation in 
m' , v', r', d', s', n', and i'. Join these points by a curve, as 
shown. 

Fig. 11 represents the development of the cylinder drawn 
in Fig. 10. Draw the horizontal line ac and the two vertical 
lines ab and cd at a distance apart equal to the diameter 1£" X 
3.1416 = 3.534", or 3-J--J" nearly. Divide ac into twelve equal 
parts, to correspond with the number of divisions in the plan 
and elevation, and draw twelve vertical lines of indefinite 
length. Make ab', ee', ff, gg', hli ', it', oo' equal to the per- 
pendicular distances of the points in, k, p, b, j, I, and i from 
the line ac in the plan respectively, and step off these same 
distances on the other side of the line oo' . Through these 
points draw the curve bo'd, and abed is the development of 
the cylinder abed in Fig. 10. 

Fig. 12 represents the end of a forked eye bolt of the 
given dimensions. The upper left-hand view is a front view, 
to the right of it the side view, and below it the bottom view. 
Draw the center lines AB, CD, EF, and GH as shown, 
giving the points o, o' , and o" . Draw the vertical lines 
aba'b' and ede'd', |" apart, li" from the latter the line efe'f, 
and |" from this the line ghg'h'. With z as a centre, located 
1J-" below CD on the line AB, describe a semicircle joining 
cd and ef. and with a radius of 1 1" describe ares terminating 
at z and X on each side of the fork. With o' as a center. 
describe a circle f" in diameter, and with a radius of \\' 
arcs, which are joined by the horizontal lines M \ \ \" apart. 
Draw the horizontal lines a'c'e'g' and b'd'fh', \\" apart, and 
dotted horizontal lines J" apart, as shown. Project the cir- 
cle with a diameter of |" upward, and join the points X and i 

by an arc haying a radius of \[". With o" as a center, 
describe circles with diameters of l V and J", and project these 



